Linear sections of the Severi variety and moduli of curves
aa r X i v : . [ m a t h . AG ] O c t LINEAR SECTIONS OF THE SEVERI VARIETY AND MODULI OF CURVES
MAKSYM FEDORCHUK
Abstract.
We study the Severi variety V d,g of plane curves of degree d and geometric genus g .Corresponding to every such variety, there is a one-parameter family of genus g stable curves whosenumerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursiveformula for the degrees of the Hodge bundle on the families in question. For d large enough, thesefamilies induce moving curves in M g . We use this to derive lower bounds for the slopes of effectivedivisors on M g . Another application of our results is to various enumerative problems on V d,g . Introduction
Statement of the problem.
Let P ( d ) ∼ = P ( d +22 ) − be the space of all plane curves of degree d .Inside it, there is a locally closed subset parameterizing nodal curves with δ nodes. Its closure V d,δ iscalled Severi variety and has been studied extensively. A generic curve in V d,δ has geometric genus g = (cid:0) d − (cid:1) − δ and, occasionally, we use V d,g instead of V d,δ . Note that we do not require curves in V d,δ to be irreducible; the closure of the locus of irreducible curves is denoted by V d,δ irr . Following[CH98a], we denote degrees of V d,δ and V d,δ irr inside P ( d ) by N d,δ and N d,δ irr , respectively.In this paper, we will be concerned with one-dimensional linear sections of V d,δ by hyperplanes ofa special form in P ( d ). Namely, consider the locus of curves passing through a fixed point p ∈ P . Itis a hyperplane in P ( d ), and is denoted by H p . Definition 1.1.
Let N = dim V d,g = 3 d + g − { p i } ≤ i ≤ N − be a set of general points in P ,then C d,δ := V d,δ ∩ H p ∩ · · · ∩ H p N − ,C d,δ irr := V d,δ irr ∩ H p ∩ · · · ∩ H p N − . ⋄ Let Y d,δ irr be the restriction of the universal family to C d,δ irr . Then its normalization ( Y d,δ irr ) ν is afamily of connected, generically smooth curves of genus g = (cid:0) d − (cid:1) − δ , and so induces a rational mapfrom C d,δ irr into M g . The natural question to ask in this situation is, what are the intersection numbersof this curve with the generators of the Picard group of M g : λ, ∆ , ∆ , . . . , ∆ ⌊ g/ ⌋ ?After a moment of reflection, we see that intersection numbers of C d,δ irr with boundary divisors canbe expressed in terms of the degrees of various Severi varieties. For example, C d,δ irr · ∆ = ( δ + 1) N d,δ +1irr . The more subtle problem is determining the degree of the Hodge bundle on C d,δ irr , i.e., the number C d,δ irr · λ, which we denote by L d,δ irr .This question is even more intriguing when the Brill-Noether number ρ ( g, , d ) = 3 d − g − C d,δ irr is a moving curve inside M g , and so the fraction C d,δ irr · ∆ L d,δ irr gives a lower bound on the slope of effective divisors on M g . The main result of this paper is the recursive formula of Theorem 1.11 that allows us to computenumbers L d,δ irr . The statement of the theorem requires several preliminary definitions which we give inthe remainder of this section.1.2. Notations and conventions.
We work over the field of complex numbers C . We denote by M g,n the coarse moduli space of stable curves of genus g with n marked points. For g ≥
2, we let C g π → M g be the “universal” curve, defined away from the codimension two locus of curves with extraautomorphisms. Then the Hodge bundle is, by definition, E := π ∗ ( ω C g /M g ). We set λ := c ( E ).Throughout the paper, δ ij stands for the Kronecker’s delta. We reserve the symbol ǫ for thegenerator of the ring of dual numbers Spec C [ ǫ ] / ( ǫ ). Given a scheme X , we denote its normalizationby X ν . The tangent space to X at a point x is denoted by T x X . The tacnode of order m is a planarcurve singularity analytically isomorphic to the singularity of y − x m = 0 at the origin. For the sakeof uniformity, we will not distinguish between a node and a tacnode of order 1.1.3. Reducible vs. irreducible curves.
For technical reasons, it is simpler to work with the vari-ety V d,δ of possibly reducible curves. Recall from Definition 1.1, that the curve C d,δ parameterizes δ -nodal curves of degree d through N − { p i } ≤ i ≤ N − in P . Consider a compo-nent of C d,δ parameterizing curves which have k irreducible components belonging to Severi varieties V d ,δ irr , . . . , V d k ,δ k irr where d = k X i =1 d i and δ = k X i =1 δ i + X ≤ i 1. Equalities (1.2) implythat P ki =1 g i − k + 1 = g . Clearly, the component in question is a union of the Segre images of theproducts k Y i =1 ( V d i ,δ i irr ∩ ( ∩ N i − δ i,r j =1 H p i,j )) = { pts } × · · · × C d r ,δ r irr × · · · × { pts } ⊂ k Y i =1 V d i ,δ i irr , for every 1 ≤ r ≤ k , and where { p i,j } = { p , . . . , p N − } (we use the shorthand { pts } to denote a finiteset of isolated points on a variety).There is a natural map j : k Y i =1 V d i ,δ i irr k Y i =1 M g i defined as the product of moduli maps. Define a line bundle Λ on Q ki =1 M g i to be the product ofpullbacks of the determinants of the Hodge bundles from the individual factors.Define now an auxiliary space M g to be a countable union of varieties Q ki =1 M g i over all k and { g i } ≤ i ≤ k satisfying P ki =1 g i − k + 1 = g . As described in the previous paragraph, M g comes withthe line bundle Λ whose first Chern class we denote by λ . By above we have a natural “moduli” map j : C d,δ M g . Note that for a given pair ( d, δ ) we need only a finite number of components of M g . Therefore, wecan define the intersection number L d,δ := j ∗ ( C d,δ ) · λ. Using this definition, we have j ∗ ( { pt } × · · · × C d r ,δ r × · · · × { pt } ) · λ = L d r ,δ r irr , where L d r ,δ r irr is defined via the map j r : C d r ,δ r irr M g r . Numbers L d,δ irr are recovered inductively from numbers L d,δ using the following formula L d,δ irr = L d,δ − X k ( d ,...,d k )( δ ,...,δ k ) k X r =1 n Y i =1 ,i = r (cid:18) d + g − d i + g i − (cid:19) N d i ,δ i irr (cid:18) d + g − d r + g r − (cid:19) L d r ,δ r irr , where the sum is taken over all k and over all ordered partitions ( d , . . . , d k ) of d , and ( δ , . . . , δ k ) of δ − P ≤ i We base our analysis of C d,δ on the degeneration approach de-veloped by Caporaso and Harris in [CH98a]. First, we recall the definition of the generalized Severivariety from ibid. Section 1.1, and the notations accompanying it.Given a sequence of non-negative integers α = ( α , α , . . . ), we define | α | = X i α i , Iα = X i iα i , I α = Y i i α i . Fx a line L ⊂ P , once and for all. Definition 1.3. For a given d and δ , consider any two sequences α = ( α , α , . . . ) and β = ( β , β , . . . )of non-negative integers such that Iα + Iβ = d. Fix a general collection of points Ω = { p i,j } ≤ j ≤ α j ⊂ L. Consider the locus of δ -nodal plane curves X of degree d that does not contain L , and such that, forthe normalization map η : X ν → X, we have η ∗ ( L ) = X i · q i,j + X i · r i,j for some | α | points q i,j and | β | points r i,j on X ν such that η ( q i,j ) = p i,j . The closure of this locus is called the generalized Severi variety and is denoted V d,δ ( α, β )(Ω) . ⋄ Remark 1.4. Since Ω is a general set, the geometry of V d,δ ( α, β )(Ω) does not depend on it. Therefore,it is customary to omit Ω from the notation. ⋄ Definition 1.5. We also define V d,δL ( α, β ) := { X ∪ L : X ∈ V d,δ ( α, β ) } ⊂ P ( d + 1) . ⋄ We recall next the main result of [CH98a]: Theorem 1.6. [CH98a, Theorem 1.2] For a general q ∈ L , we have the following equality of cycles V d,δ ( α, β )(Ω) ∩ H q = X k kV d,δ ( α + e k , β − e k )(Ω ∪ { q } )+ X I β ′ − β (cid:18) β ′ β (cid:19) V d − ,δ ′ L ( α ′ , β ′ )(Ω ′ ) , (1.7) where the second sum is taken over all triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − , and over all sets of points Ω ′ = { p ′ i,j } ≤ j ≤ α ′ i ⊂ Ω . We refer to components of H q := V d,δ ( α, β )(Ω) ∩ H q appearing on the first line of Equation (1.7)as Type I components . A generic point of a Type I component does not contain the line L . Theremaining components of H q are called Type II components and parameterize curves containing theline L .An immediate corollary of Theorem 1.6 is Theorem 1.8. [CH98a, Theorem 1.1] The degrees N d,δ ( α, β ) of the Severi varieties V d,δ ( α, β ) satisfythe recursion N d,δ ( α, β ) = X k kN d,δ ( α + e k , β − e k )+ X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) N d − ,δ ′ ( α ′ , β ′ ) , where the second sum is taken over all triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − . Throughout the paper we work with linear sections of V d,δ ( α, β ). We introduce the followingnotations. Definition 1.9. Let N := dim V d,δ ( α, β ) and { p i } ≤ i ≤ N − be the set of N − P .Set S d,δ ( α, β ) := V d,δ ( α, β ) ∩ H p ∩ · · · ∩ H p N − and C d,δ ( α, β ) := V d,δ ( α, β ) ∩ H p ∩ · · · ∩ H p N − . ⋄ Note that S d,δ ( α, β ) is a surface and C d,δ ( α, β ) is a divisor on it. Definition 1.10. For g = (cid:0) d − (cid:1) − δ , we let j to be the induced rational map j : C d,δ ( α, β ) M g , where M g is the scheme described in Section 1.3. We set L d,δ ( α, β ) := j ∗ ( C d,δ ( α, β )) · λ. ⋄ The strategy for calculating numbers L d,δ is now clear. Using the degeneration of Theorem 1.6, weproduce a recurrence relation among numbers L d,δ ( α, β ) paralleling that of Theorem 1.8. The maintheorem of this paper is Theorem 1.11. The numbers L d,δ ( α, β ) satisfy the recursion L d,δ ( α, β ) = X k kL d,δ ( α + e k , β − e k )+ X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) L d − ,δ ′ ( α ′ , β ′ )+ 112 X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) · (cid:18)X k ( β ′ k − β k )( k − (cid:19) · N d − ,δ ′ ( α ′ , β ′ ) ; where the second sum is taken over all triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − and thethird sum is taken over all triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − . It follows from Equation (1.7) that only varieties V d − ,δ ′ L ( α ′ , β ′ )(Ω ′ ) with α ′ ≤ α, β ′ ≥ β and δ ′ ≤ δ appear withthe non-zero coefficient. Intuitively, the first and the second line of the recursion do not require an explanation. GivenTheorem 1.6, they are, at least, expected to appear. The third line of the recursion is of differentnature. It arises from an extra complication that did not appear in the analysis of [CH98a]. We pausehere to describe it.The degree is an intrinsic property of the Severi variety, as it comes naturally with the definition.On the other hand, the first Chern class of the Hodge bundle on C d,δ ( α, β ) is defined in terms ofan extra structure that we put on the Severi variety, namely, the moduli map to M g . Moreover,the moduli map is not rational. It would not pose much difficulty if we were considering C d,δ ( α, β )alone, as the map would then naturally extend to the regular map (at least after the normalization).However, our approach is to degenerate C d,δ ( α, β ) to a union of linear sections of “simpler” generalizedSeveri varieties. The total space S of the degeneration is thus two-dimensional, and we no longer canexpect the moduli map to extend. To resolve the moduli map, we have to blow-up S . The exceptionaldivisors of the blow-up will then contribute to the calculation. These contributions appear on thethird line of the recursion in Theorem 1.11.Yet another way to think of the problem is to recall that we have the Kontsevich moduli space ofstable maps M g, ( P , d ) that fits into the diagram M g, ( P , d ) π (cid:15) (cid:15) / / M g V d,g ttttt (1.12)The projection π maps the so called “main component” of M g, ( P , d ) birationally onto V d,g . Thespace of stable maps has the advantage that the moduli map to M g is a well-defined morphism.The difficulty that arises, if one wants to work with the stable maps, is the existence of componentsof M g, ( P , d ) parameterizing maps that contract components of positive genus. Moreover, thesecomponents of M g, ( P , d ) have wrong dimension.The moduli map from S to M g will be undefined precisely at the points corresponding to thestable maps contracting components of positive genus. To resolve the map, we are thus required tounderstand the proper transform of S inside M g, ( P , d ).Finally, even though we do not use the language of stable maps in this paper, the above discussionserves as a motivation for much of what follows.1.5. Structure of the paper: In Chapter 2, we explain the notions of λ -indeterminacy and discrep-ancy that we encounter when working with two-dimensional families of not necessarily stable curves.In Chapter 3, we describe explicitly where the indeterminacy occurs along a special hyperplane sec-tion of S d,δ ( α, β ). In Chapter 4, we reduce the discrepancy calculation on S d,δ ( α, β ) to a discrepancycalculation on a surface in the product of deformation spaces of several tacnodes. We develop a theorywhich allows to do this local calculation in Chapters 5 and 6, and perform the calculation in Chapter7. In the final chapter of the paper, we give examples of enumerative problems that can be solvedusing our Theorem 1.11.1.6. Acknowledgments: We are grateful to Joe Harris for numerous discussions, advice, and forintroducing us to this problem. We we would like to thank Ethan Cotterill, for reading the preliminarydraft and providing valuable suggestions, and Anatoly Preygel, whose Python script served as a basisfor the program implementing the recursion of our Theorem 1.11.2. Remarks on the intersection theory In this chapter, we recall some generalities on the coarse moduli space M g of stable curves of genus g . The reference for the material presented here is [HM98, Chapter 3]. The code is available upon request. Families of curves over one-dimensional bases. Consider a proper family X → C of curvesover a one-dimensional irreducible base C . Suppose that a generic fiber is a nodal curve of arithmeticgenus g . Then there is a finite surjective base change f : C ′ → C , and a family π : Y → C ′ of stablecurves of genus g such that Y is birational to X × C C ′ . Colloquially, Y → C ′ is a stable reduction of X → C . The Hodge bundle E Y on C ′ is defined by E Y := π ∗ ( ω Y /C ′ ) . It is a pullback of the Hodge bundle on M g under the natural morphism C ′ → M g induced by thefamily Y → C ′ . Without performing a stable reduction, we still have a natural rational map from C to M g : j : C M g . Definition 2.1. The degree of λ on X /C , denoted by λ X /C , is the intersection number j ∗ ( C ) · λ .It is also equal to c ( E Y ) / deg f . ⋄ Note that λ X /C depends only on the geometry of the family at a generic point of C . If the family X → C is understood, we use the shorthand λ C to denote the degree of λ on X /C .The following two lemmas follow from definitions. Lemma 2.2. Suppose X → C is a flat proper family of curves with a nodal generic fiber. If X = X ∪ X , a union of two families, then λ X /C = λ X /C + λ X /C . Lemma 2.3. Suppose X → C is a flat proper family of curves of arithmetic genus g . Suppose,moreover, that a generic fiber of X has δ nodes and no other singularities. Then the normalization X ν → C is a proper family, whose generic fiber is a smooth curve of genus g − δ , and we have λ X /C = λ X ν /C . Families of curves over two-dimensional bases. Consider now a flat proper family X → B over a two-dimensional irreducible base B whose generic fiber is a stable curve of arithmetic genus g .We have an induced rational map j : B M g , which we call the moduli map . The locus U of points b such that X b is stable is an open subset of B .Note that j is regular on U . Replace now B by its normalization, and denote by X its own pullbackto the normalization, and by U its own preimage.By properness of M g , the moduli map j is defined away from a finite set of points of B . We callthem points of indeterminacy of the moduli map, and denote the set of such points by Indet( X /B );or Indet( B ) if the family X is understood. Set W = B r Indet( B ). Then U is a subset of W , and theinclusion can be proper. For example, suppose b ∈ B r U is such that the isomorphism class of thestable limit of any one-parameter family with the center at b does not depend on the family. Then j extends to a regular map in a neighborhood of b , even though X b is not stable.The resolution of j is a proper birational map π : ˆ B → B , restricting to the isomorphism on U ,together with a regular map ˆ : ˆ B → M g such that the following diagram commutes:ˆ B π (cid:15) (cid:15) ˆ (cid:31) (cid:31) @@@@@@@@ B j / / ___ M g (2.4)By the Zariski’s Main Theorem, π : ˆ B → B has connected fibers. We note that it is possible to testwhether b ∈ B will be a point of indeterminacy without passing to the normalization. Suppose thatthe isomorphism class of the stable limit of any one-parameter family with the center at b belongs toa discrete set. Then, after the normalization, the moduli map is defined at b . Indeed, the finitenessassumption implies that π − ( b ) is a finite set, and so π must be an isomorphism at b . When C is smooth, the map extends to a regular morphism. For every curve C ⊂ B , we can define the number λ C by restricting j to C and setting λ C := j ∗ ( C ) · λ. Note that if X × B C is a family with a nodal generic fiber, then λ C = λ X × B C/C .For every Q -Cartier divisor C ⊂ B , we define another closely related number. Definition 2.5. The number ( λ · C ) B := ˆ ∗ ( λ ) · π ∗ ( C )is called the λ -degree of C on B . This also equals to λ C ′ for any divisor C ′ that does not pass throughIndet( B ) and is linearly equivalent to C . ⋄ For a Q -Cartier divisor C , the pullback is defined and we have π ∗ ( C ) = π − ∗ C + E C , where E C is some linear combination of exceptional divisors of π . We define the λ -discrepancy, orsimply discrepancy, along C to be the number Discrep λB ( C ) := ( λ · C ) B − λ C . For b ∈ Indet( B ), we define Discrep λb ( C ) := X E · λ where the sum is taken over all exceptional divisors E mapping to b .Note that since M g is only a coarse moduli space, there might not be a family of stable curvesover ˆ B that induces the map ˆ . However, by [HM98, Lemma 3.89], there is a finite order base changeˆˆ B f −→ ˆ B and a family of stable curves Y → ˆˆ B such that ˆ ◦ f : ˆˆ B → M g is induced by the family Y .Moreover, we have X × U f − ( U ) ∼ = Y f − ( U ) . We set ˆ π := π ◦ f . By abuse of terminology, we call any such family, Y → ˆˆ B , a stable reduction of X → B (cf. [HM98, Corollary 3.96]).Given a Q-Cartier divisor C ⊂ B , we define its strict transform on ˆˆ B to be C := ˆ π − ( C ∩ W ). Wedefine the exceptional part of C to be ˆ E C := ˆ π ∗ ( C ) − C. Then the λ -discrepancy along C on B equalsto 1deg f ( ˆ E C · λ ) . Note that ˆ E C · λ is defined unambiguously. It is the degree of the Hodge bundle corresponding to thefamily of stable curves, Y ˆ E C → ˆ E C . Lemma 2.6. If b ∈ B is such that the geometric genus of the fiber X b is g , then b / ∈ Indet ( B ) .Proof. Suppose π : ˆ B → B is a minimal resolution of the moduli map, and E ⊂ ˆ B is an exceptionaldivisor mapping to b . After a finite base change, we can assume that we have a family Y of stablecurves over ˆ B . The fibers of Y over E are stable curves of arithmetic genus g that map to X b . Sincethe normalization ( X b ) ν of X b has geometric genus g , we conclude that all fibers of Y over E mustbe isomorphic to ( X b ) ν . Hence, E maps to a point in M g and so cannot be an exceptional divisor of π . (cid:3) Given two linearly equivalent divisors C and C ′ on B , we have, by definition, ( λ · C ) B = ( λ · C ′ ) B .If, moreover, C ′ lies in U , we can compute λ C in terms of λ C ′ and the discrepancy Discrep λS ( C ). Thissimple observation is important because the discrepancy along C depends only on the geometry ofthe family X → B in the analytic neighborhood of B around Indet( B ). Therefore, to compute thediscrepancy we can work locally around each point of indeterminacy. 3. The degeneration and points of indeterminacy Consider the surface S d,δ ( α, β ) = V d,δ ( α, β ) ∩ H p ∩ · · · ∩ H p N − and a restriction of the universal curve to S d,δ ( α, β ) which we denote by Y := Y d,δ ( α, β ). A genericfiber of Y is a δ -nodal curve of geometric genus g = (cid:0) d − (cid:1) − δ . By discussion in Chapter 2, we havean induced rational moduli map S d,δ ( α, β ) M g .Let p N − be a general point in P . Then by Definition 1.9, S d,δ ( α, β ) ∩ H p N − = C d,δ ( α, β ) . (3.1)Suppose now q is a general point on the line L . Then by the Theorem 1.6, we have S d,δ ( α, β ) ∩ H q = X k kC d,δ ( α + e k , β − e k )+ X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) C d − ,δ ′ L ( α ′ , β ′ ) . (3.2)Equalities (3.1) and (3.2) show that there is a linear equivalence C d,δ ( α, β ) ∼ X k kC d,δ ( α + e k , β − e k )+ X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) C d − ,δ ′ L ( α ′ , β ′ )(3.3)of divisors on S d,δ ( α, β ).By results of [CH98a], the stable limit of a generic arc with the center at a generic point [ C ] = [ Y ∪ L ]of C d − ,δ ′ L ( α ′ , β ′ ) is a nodal curve, which is a partial normalization of C , equal to a union of the line L and the normalization of the curve X . Recalling the discussion of Chapter 2, we have the followingresult. Corollary 3.4. After the normalization of S d,δ ( α, β ) , the moduli map is defined at a generic pointof any Type II component. Invoking Lemmas 2.2 and 2.3, we have λ C d − ,δ ′ L ( α ′ ,β ′ ) = L d − ,δ ′ ( α ′ , β ′ ) . Finally, we have, by the definition, L d,δ ( α, β ) + Discrep λS d,δ ( α,β ) ( C d,δ ( α, β ))= X k kL d,δ ( α + e k , β − e k ) + X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) L d − ,δ ′ ( α ′ , β ′ ) + Discrep λS d,δ ( α,β ) ( H q ) . (3.5)Therefore to prove Theorem 1.11, we need to understand the points of indeterminacy Indet( S d,δ ( α, β ))of the moduli map j and the λ -discrepancies along C d,δ ( α, β ) and H q .3.1. Indeterminacy Points. In this section we describe the points of indeterminacy that occur alonglinear sections H p N − and H q . The main tools in our analysis are nodal reduction for curves and thefollowing fundamental dimension-theoretic result on the deformations of plane curves: Lemma 3.6. [CH98a, Corollary 2.7] Fix a subset Ω of general points on L ⊂ P . Let V be a locallyclosed irreducible subvariety of P ( d ) and X a generic point of V . Let π : X ν → X be the normalizationmap and e := card( X ∩ ( L r Ω)) . Then dim V ≤ d + g − e. (3.7) Moreover, if equality holds and card( π − ( L r Ω)) = e , then V is a dense open subset of a generalizedSeveri variety. The following is the main result of this chapter. Proposition 3.8. The generic hyperplane section C d,δ ( α, β ) does not contain points of indeterminacy.The points of indeterminacy that lie on H q are V d,δ ′ ( α ′ , β ′ ) ∩ H p ∩ · · · ∩ H p N − , where ( δ ′ , α ′ , β ′ ) satisfy | β ′ − β | + δ − δ = d − , and β ′ i > for some i ≥ . (3.9) Remark 3.10. The points of indeterminacy on H q can also be described as points of V d,δ ′ ( α ′ , β ′ − e i − e j + e i + j ) ∩ H p ∩ · · · ∩ H p N − , where V d,δ ′ ( α ′ , β ′ ) is a Type II component of H q . ⋄ Nodal reduction: In our analysis of indeterminacy points on H q , we will often need to under-stand the stable limits of one-parameter families inside S d,δ ( α, β ). We use the approach of Section3 of [CH98a] in what follows. Let ∆ be an arc in S d,δ ( α, β ) with the center at [ C ] = [ Y ∪ L ] ∈ C d − ,δ ′ L ( α ′ , β ′ ) and Y = Y d,δ ( α, β ) × S ∆ the restriction of the universal family.It is a standard result that we may perform a nodal reduction of Y → ∆ to obtain a flat family Z → ∆ of generically smooth genus g curves satisfying the following conditions:(1) The total space Z is smooth and there is a map η : Z → P . (2) The central fiber Z is nodal.(3) Z decomposes as Z = Y ∪ P, where Y is a strict transform of Y , and P is a union of components mapping to L .(4) There is a multisection F = P Q i,j such that η ( Q i,j ) = p i,j , and a multisection V = P R i,j such that η ∗ ( L ) = X i,j i · Q i,j + X i,j i · R i,j + P , where P is a divisor supported on P . Z ⊂ Z (cid:15) (cid:15) η / / (cid:15) (cid:15) P [ C ] ∈ ∆We consider a decomposition P = c X i =1 P ′ i + c X i =1 P ′′ i of P into connected components such that P ′ i ’s are not contracted by η and P ′′ i ’s are contracted by η .Every contracted component P ′′ i has to meet either V or F . We let the union of these to be P ′′ v and P ′′ f , respectively. Denote the number of connected components of P ′′ v and P ′′ f by v and f , respectively. Proof of Proposition 3.8: First, we consider H p N − . Note that H p N − intersects any Type II component at a generic point ofthe component. By Corollary 3.4, the moduli map is defined at these points. All other points of H p N − correspond to curves not containing the line L , and we next consider only such points. By Lemma3.6, the only points on H p N − where the geometric genus drops are ( δ + 1)-nodal curves. The modulimap is clearly defined at these points. By Lemma 2.6 there are no other points of indeterminacy. Indeterminacies on Type I components: Consider a curve C on a Type I component C d,δ ( α + e k , β − e k ) of S d,δ ( α, β ) ∩ H q . By Lemma 2.6,a point of indeterminacy can occur only when the geometric genus drops. If C does not contain a line,by Lemma 3.6, genus can drop by at most 1. In this case we must have card( C ∩ ( L r Ω)) = | β − e k | ,and the inequality in (3.7) becomes equality. Therefore, C is a ( δ + 1)-nodal curve, smooth along L .0We conclude that potential points of indeterminacy have to contain the line L . By Theorem 1.6, theseare points in V d − ,δ ′ ( α ′ , β ′ ) such that | β ′ − ( β − e k ) | + δ − δ ′ = d − ⇔ | β ′ − β | + δ − δ ′ = d − . Indeterminacies on Type II components: Suppose [ C ] ∈ C d − ,δ ′ L ( α ′ , β ′ ) is a point of indeterminacy. By definition, C = Y ∪ L where Y is in C d − ,δ ′ ( α ′ , β ′ ), for some ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − . (3.11)First, suppose that Y does not contain L . Let card( Y ∩ ( L r Ω)) = e . Here e ≤ | β ′ | . Then byLemma 3.6, 2( d − 1) + g ( Y ) + e − ≥ dim V d,δ ( α, β ) − d + g + | β | − − . (3.12)Equivalently, ( g ( Y ) − ( g − | β ′ − β | + 1)) + ( e − | β ′ | ) ≥ − 1. On the other hand, neither of the summandson the left hand side is greater than 0. If g ( Y ) = g − | β ′ − β | + 1 and e = | β ′ | , then C is not a point ofindeterminacy. Indeed, in this case, C cannot be an image of a stable map of genus g that contractsa component of a positive genus.Suppose g ( Y ) = g − | β ′ − β | . Then e = | β ′ | which forces all inequalities in (3.7) to be equalitiesand Y to be unibranch at every point of L r Ω. Therefore, Y is a generic point of V d − ,δ ′ +1 ( α ′ , β ′ ),a generalized Severi variety satisfying condition (3.9).Consider now the case when e = | β ′ | − g ( Y ) = g − | β ′ − β | + 1. If Y is unibranch at everypoint of ( L r Ω), then by Lemma 3.6, the curve Y belongs to the Severi variety V d − ,δ ′ ( α ′ , β ′′ ), with | β ′′ | = | β ′ | − 1. Informally, we see two of the new points points of tangency on V d,δ ( α, β ) coalesce. Ina such situation, the newly formed tacnode of order m on Y ∪ L is a limit of only m − S d,δ ( α, β ).Suppose Y is not unibranch along L r Ω. We consider a nodal reduction of a generic one-parameterfamily in S d,δ ( α, β ) with the center at [ C ]. We use notations of Section 3.2 throughout. Since [ C ]contains L with multiplicity 1, we have c = 1. Let β is the number of sections in V meeting Y . Weset β = | β | − β . Note that β + card(( P ′ + P ′′ v ) ∩ Y ) = card( π − ( L r Ω)) ≥ e + 1 = | β ′ | . Also, card( P ′′ f ∩ Y ) ≥ f and v ≤ β . Putting everything together we have the following inequalities g = p a ( Z ) ≥ g ( Y ) + (1 − c − c ) + card( P ′ ∩ Y ) + card( P ′′ ∩ Y ) − g ( Y ) − ( v + f ) + card( P ′ ∩ Y ) + card( P ′′ ∩ Y ) − g + (card( P ′′ f ∩ Y ) − f ) + ( β + card( P ′ ∩ Y ) + card( P ′′ v ∩ Y ) − | β ′ | ) + ( β − v ) ≥ g. We conclude that:(1) All connected components of P have arithmetic genus 0.(2) card( P ′′ f ∩ Y ) = f .(3) card( π − ( L r Ω)) = e + 1 = | β ′ | .(4) Connected component of P ′′ v are in one to one correspondence with β sections of V notmeeting Y .It follows from (1) and (2) that P ′′ f is empty. Also, from (1) any component of P ′′ v has to meet Y in at least two points, and from (3) it follows that v is at most 1. In every case, the stable limit isone of the finitely many curves. We can have either v = 1, with the only component of P ′′ v being a P meeting Y in two points and meeting one of the sections R i,j . In this case, P ′ meets Y in pointswhich all map to different points on L . The other possibility is that P ′′ is empty and P ′ meets Y ina set of points among which there are two that map to the same point on L . The first possibility ispresented in Figure 1, where the curve on the left is a central fiber of the nodal reduction, the curvein the center is its stabilization. The curve on the right is the planar image. The second possibility ispresented in Figure 2. It follows that [ C ] is not a point of indeterminacy.1 LP ′′ R i,j P ′ P ′ Y η ( q i,j ) R i,j YY Figure 1. Y Y LP ′ YP ′ Figure 2. Remaining case: Y contains L . We set Y = X ∪ L where Y is a general point of the Severivariety V d,δ ′′ ( α ′′ , β ′′ ) with | β ′′ − β ′ | + δ − δ ′′ = d − 2. Together with (3.11), this is equivalent to g ( X ) = g − | β ′′ − β | + 2 . Following the notations of the Section 3.2, we consider the nodal reduction of a generic family in S d,δ ( α, β ) with the center at [ C ].Note that c ≤ P ′′ ∩ X ) ≥ c since Z is connected. We have g = p a ( Z ) ≥ g ( X ) + (1 − c − c ) + ( P ′ ∩ X ) + ( P ′′ ∩ X ) − ≥ g − | β ′′ − β | + ( P ′ ∩ X ) + (2 − c ) + (( P ′′ ∩ Y ) − c ) ≥ g. This is possible only if all the inequalities are equalities. We draw the following conclusions.(1) All connected components of P have arithmetic genus 0.(2) P ′′ is empty.(3) There are two components, P ′ and P ′ , of P ′ . Each is a tree of rational curves that is mappedwith degree 1 onto L .We conclude that the stable reduction of Z looks like the curve in Figure 3. In particular, thereare finitely many possible stable limits of one-parameter families with the center at [ C ], and so [ C ] isnot a point of indeterminacy. P ′ P ′ X LX Figure 3. The proof of Proposition 3.8 is finished.2We conclude this chapter with the restatement of Equation (3.5): L d,δ ( α, β ) = X k kL d,δ ( α + e k , β − e k )+ X I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) L d − ,δ ′ ( α ′ , β ′ ) + Discrep λS d,δ ( α,β ) ( H q ) . (3.13) 4. Local Geometry of Y d,δ ( α, β )In this chapter, we describe the geometry of the family Y d,δ ( α, β ) → S d,δ ( α, β ). Note that theonly singularities, besides nodes, that appear on the curves corresponding to generic points of TypeII components, and on the curves corresponding to the points of indeterminacy, are higher-ordertacnodes. Therefore the description of the local geometry of Y d,δ ( α, β ) invariably invokes versaldeformation spaces of (arbitrary order) tacnodes. We recall the necessary definitions in what follows.4.1. Versal deformation space of a tacnode. Recall that the versal deformation space of the m th -order tacnode y − x m = 0 is T ∼ = C m − . We let the coordinates on T be ( a , . . . , a m ). Thenthe miniversal family Y → T can be defined by the equation y = ( x m + a x m − + a x m − + · · · + a m ) + a m +1 x m − + · · · + a m − x + a m (4.1)inside T × Spec C [ x, y ].For a = ( a , . . . , a m ) ∈ T , we write Ψ( a , . . . , a m )( x ), or Ψ a ( x ), to denote the polynomial on theright hand side of Equation (4.1). We use Ψ a ( x, z ) to denote the homogenization of Ψ a ( x ).Let D ∼ = C m − be the space of monic polynomials of degree 2 m with a trivial x m − coefficient.Any polynomial in D can represented as a sum of the square of a monic polynomial of degree m and apolynomial of degree m − 1. In fact, the map Ψ : T → D sending a point a ∈ T to a polynomial Ψ a ( x )is an isomorphism, since its Jacobian is an upper-triangular matrix with non-zero complex numbersalong the diagonal. Definition 4.2. A combinatorial type of a polynomial Ψ of degree d is an r -tuple ( m , . . . , m r ) ofmultiplicities of distinct roots of Ψ. Here r is the number of distinct roots of Ψ. ⋄ The deformation space T has a natural geometric stratification given by the combinatorial type ofthe polynomial Ψ a ( x ). Definition 4.3. We denote the stratum of deformations of type ( m , . . . , m r ) by∆ { m , . . . , m r } ;and denote its closure by ∆ { m , . . . , m r } . ⋄ Remark 4.4. For a ∈ ∆ { m , . . . , m r } , the singularities of the fiber y = Ψ a ( x ) are double points y = x m i , therefore ∆ { m , . . . , m r } is an equisingular stratum. ⋄ We also define ∆ r := ∆ { , , . . . , | {z } r , , . . . , | {z } m − r } to be the closure of the locus of r -nodal curves.We will distinguish the hyperplane H m inside T defined by a m = 0. Under the identification ofthe tangent space T T with the space Def ( y = x m ) of the first-order deformations of the tacnode, T H corresponds to the first order deformations of y = x m vanishing at (0 , ∈ C .34.2. Multiple tacnodes. Let m = ( m , . . . , m n ) be a sequence of positive integer numbers. For1 ≤ i ≤ n , let T ( i ) ∼ = C m i − be the versal deformation space of the tacnode y − x m i = 0, and let( a i, , . . . , a i, m ) be coordinates on T ( i ). We set T := n Y i =1 T ( i ) ∼ = C P i m i ) − n to be the product of these deformation spaces.By analogy with a single tacnode case, we define the following loci inside T:∆ m := n Y i =1 ∆ i,m i , ∆ m − := n Y i =1 ∆ i,m i − , ∆ m − − e j := n Y i =1 ∆ i,m i − − δ i,j , ≤ j ≤ n. Geometry at the generic point of Type II component. We recall the description, givenin [CH98a, Section 4.4], of the geometry of S d,δ ( α, β ) around a generic point of a Type II componentof H q .Suppose that [ X ] = [ Y ∪ L ] ∈ C d − ,δ ′ L ( α ′ , β ′ ) is a generic point. Here, [ Y ] is a generic point of C d − ,δ ′ ( α ′ , β ′ ). The curve X has | α ′ | + | β ′ | tacnodes, at the points of contact of Y with L , of which | α ′ | are in the set Ω ⊂ L . Among the “moving” | β ′ | points of contact, exactly | β | are the limits of“moving” points of tangency in the nearby fibers. We call the corresponding tacnodes of X “old” .The rest of the tacnodes correspond to “new” points of tangency of Y with L . Set β ′′ := β ′ − β and n := | β ′′ | . Denote the new tacnodes of X by y , . . . , y n , and their multiplicities by m , . . . , m n .Consider the Severi variety V d,δ ′′ ( α, β ), where δ ′′ = δ − ( Iβ ′′ − n ). Inside it, we define an opensubvariety V consisting of the deformations of X = Y ∪ L in V d,δ ′′ ( α, β ) satisfying the followingconditions:(1) Tangencies at Ω are preserved.(2) The deformations of the | β | “old” tacnodes preserve two branches: the deformation of Y and L , respectively.(3) The deformation of Y near an “old” tacnode of order i remains tangent (at an unspecifiedpoint) to the line L with the multiplicity i .The subvariety V is a relaxed Severi variety in the sense of [CH98a, Proposition 4.8]. Then, in theneighborhood of X , the variety V d,δ ( α, β ) is a closure of deformations of X inside V such that everytacnode of order y i deforms to m i − φ : V → T := n Y i =1 Def( y i , X ) . Let { a i,j } ≤ j ≤ m i be the coordinates on T i := Def( y i , X ) as described in Section 4.1. Then summa-rizing the results of [CH98a, Section 4.4], and Lemma 4.9 [CH98a] in particular, we have:(1) V is smooth.(2) φ is smooth at [ X ].(3) W := φ ( V ) contains ∆ m and is smooth of dimension P ( m i − 1) + 1.(4) The tangent space to W at the origin is not contained in the union of hyperplanes a i, m i = 0.(5) W ∩ ∆ m − = ∆ m ∪ Γ m , where Γ m is a curve intersecting ∆ m at the origin with the multiplicity M = n Y i =1 m i . (6) V d,δ ( α, β ) = φ − (Γ m ) and V d − ,δ ′ L ( α ′ , β ′ ) = φ − (0). This does not mean that an old tacnode is a limit of tacnodes in the nearby fibers. Y → S d,δ ( α, β )at a generic point [ X ] = [ Y ∪ L ] of a Type II component C d − ,δ ′ L ( α ′ , β ′ ) involves no blow-ups. Indeed,since the “new” tacnode y i , of order m i , is a limit of m i − X ] will have a node lying over y i . All othersingularities are resolved in the process of the stable reduction. We conclude that the stable limit willbe a nodal union of Y ν with the line L at the points lying over y i ’s.4.4. Geometry at the point of indeterminacy. We retain the notations of the previous section.Suppose [ X ] = [ Y ∪ L ] is a point of indeterminacy, where Y is a generic point of V d − ,δ ′ ( α ′ , β ′ )satisfying | β ′ − β | + δ − δ ′ = d − 2. Note that there are precisely (cid:18) αα ′ (cid:19) N d − ,δ ′ ( α ′ , β ′ )(4.5)points of indeterminacy corresponding to the triple ( δ ′ , α ′ , β ′ ).First, we note that S d,δ ( α, β ) has several analytic branches in the neighborhood of [ X ]. Eachbranch is specified by the choice of tacnodes of [ X ] which are the limits of the | β | moving points oftangency in the nearby fibers. There are (cid:0) β ′ β (cid:1) such branches. Choose one of them, and denote itby S . Let y , . . . y n be the remaining “new” tacnodes of X , of order m , . . . , m n , respectively (here, n = | β ′ − β | ). Let V be the relaxed Severi variety of the previous section, but now defined inside theSeveri variety V d,δ ′′ ( α, β ), where δ ′′ = δ − ( Iβ ′′ − n ) + 1. Then, by [CH98a, Lemma 4.9], the map φ : V → T := n Y i =1 Def( y i , X ) , satisfies the following conditions:(1) V is smooth.(2) φ is smooth at [ X ].(3) W := φ ( V ) contains ∆ m and is smooth of dimension P ( m i − 1) + 1.(4) The tangent space to W at the origin is not contained in the union of hyperplanes a i, m i = 0.However, the geometry of V d,δ ( α, β ) in the neighborhood of [ X ] in V is more complicated. We proceednow to describe it.Speaking colloquially, we see only P ni =1 ( m i − − y , . . . , y n . Hence,there is a tacnode y i that is the limit of only m i − y i ( i = i )is the limit of m i − ≤ i ≤ n , the intersection W ∩ ∆ m − − e i = ∆ m ∪ S i , where S i := ( W ∩ ∆ m − − e i ) r ∆ m is residual to ∆ m in the intersection. We set S m := n [ i =1 S i . Then S is identified with S m via the map φ . The hyperplane section H q of S is identified with S m ∩ ∆ m .Symbolically, S = φ − ( S m ) and S ∩ H q = φ − ( S m ∩ ∆ m ) . By the definition of φ , locally around the point y i in the central fiber X , the family Y is isomorphicto the pullback of the miniversal family over Def( y i , X ).We thus reduce the calculation of the discrepancy along H q on the branch S of S d,δ ( α, β ) to thecalculation of the discrepancy along the divisor ∆ m on the surface S m inside the product of the versaldeformation spaces of tacnodes. Recalling that the number of the points of indeterminacy of the type( δ ′ , α ′ , β ′ ) is given by (4.5), and that there are (cid:0) β ′ β (cid:1) choices for the branch S , we conclude that Discrep λS d,δ ( α,β ) ( H q ) = X (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) N d − ,δ ′ ( α ′ , β ′ ) · Discrep λS m (∆ m ) , (4.6)5where the sum is taken over the triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − m = { , . . . , | {z } β ′ − β , , . . . , | {z } β ′ − β , . . . } . Geometry of the versal deformation space of tacnode In the previous chapter, we saw that the local geometry of the Severi variety is reflected in thegeometry of certain loci in the product of the versal deformation spaces of tacnodes. In this chapter,we lay out the framework for studying the deformation space of a single tacnode. We obtain severalresults which elucidate the geometry of the miniversal family. They will be generalized to multipletacnodes in the following chapter, and used to in the calculation of discrepancies.5.1. Alterations. We use the notations of Section 4.1. Recall that the versal deformation space ofthe singularity y = x m is T = Spec C [ a , . . . , a m ]. The miniversal family Y over T is defined by y = Ψ a ( x ) = ( x m + a x m − + a x m − + · · · + a m ) + a m +1 x m − + · · · + a m − x + a m (5.1)inside T × Spec C [ x, y ].We make a base change π : T ′ = Spec C [ b , b , . . . , b m ] → T defined by a i = b ii . For µ := µ × µ × · · · × µ m , where µ r is the cyclic group of r th roots of unity, we have T = T ′ //µ .We let π : T ′′ → T ′ to be the blow-up of T ′ at the origin, whose exceptional divisor we denote by E , and set π := π ◦ π : T ′′ → T. Denote by Y ′′ ⊂ T ′′ × Spec C [ x, y ] the pullback of Y to T ′′ . Let I E be the ideal sheaf of E on T ′′ , andconsider the ideal sheaf I := (( I E , x ) m , y )on T ′′ × Spec C [ x, y ].First, note that Bl I ( T ′′ × Spec C [ x, y ]) → T ′′ is a family of surfaces with fibers over T ′′ r E beingaffine planes C , and fibers over the points in the exceptional divisor E being the union of Bl ( x m ,y ) C and P (1 , , m ) = Proj C [ x, y, z ]. Here, z stands for a local generator of I E and C [ x, y, z ] is graded withdeg x = deg z = 1 and deg y = m . Finally, we denote Z := Bl I Y ′′ and F : Z → T ′′ . The family F : Z → T ′′ can be seen as a first step towards the stable reduction of the miniversalfamily Y → T . The following paragraphs make this statement more precise.Consider a usual open cover of the blow-up T ′′ by the affine charts D ( b i ) T ′′ := Spec C [ b /b i , b /b i , . . . , b i , . . . , b m /b i ] . Then, over D ( b i ) T ′′ , the family Z has equation y = Ψ (cid:18)(cid:18) b b i (cid:19) , . . . , (cid:18) b m b i (cid:19) m (cid:19) ( x, b i ) . The restriction E ( b i ) of the exceptional divisor to D ( b i ) T ′′ is given by equation b i = 0, and therestriction of Z to E ( b i ) is Z E := Z × T ′′ E ( b i ) = S ∪ T , where S is defined by { ( y/x m ) = 1 } ⊂ E ( b i ) × Bl ( x m ,y ) C and T is defined by { y = Ψ (cid:18)(cid:18) b b i (cid:19) , . . . , (cid:18) b m b i (cid:19) m (cid:19) ( x, z ) } ⊂ E ( b i ) × P (1 , , m ) . In words, over E ( b i ) , the family Z E has two components. The first component, S , is a trivial familywith the normalization of y = x m as its fibers. The second component, T , is a family of divisors, not6passing through the vertex, in the linear series |O (2) | on P (1 , , m ) . We call fibers of T tails . Notethat all such tails are hyperelliptic curves of arithmetic genus m with 2 distinguished mark pointsdefined by z = 0. The marked points are the points of intersection with S and are exchanged by thehyperelliptic involution y 7→ − y .We observe that singularities appearing on the hyperelliptic tails are planar double points y = x n with n < m , and so are strictly better singularities than the tacnode y = x m . Hence, after analteration of the base and an appropriate blow-up of the total family, we arrived at the family of curveswith milder singularities. Conceivably, by repeating the procedure for all singularities, we would arriveat the family of the stable curves. However, for our purposes, the family F : Z → T ′′ will suffice. Caution: Even though the family S is a trivial family when restricted to affine charts E ( b i ) , it isnot trivial globally.5.2. Local charts. Note that all blow-ups in the previous subsection were made with µ -invariantcenters. Hence, the action of µ extends to Z making the morphism F equivariant. For every 2 ≤ i ≤ m , we consider a quotient of F : Z → T ′′ by the natural action of˜ µ i := µ × · · · × ˆ µ i × · · · × µ m . We set Z i := Z // ˜ µ i and T i := T ′′ // ˜ µ i . The quotient morphism F i : Z i → T i is nothing else thanthe weighted blow-up of T with weights (2 , , . . . , m ), followed by a base change a i = b ii . We define T i to be the quotient T // ˜ µ i of the tails component. By abuse of notation, we also use E to denote theexceptional divisor of F i . If W is a subvariety of T , we denote by W E its exceptional divisor in T i .Note that the action of ˜ µ i is free on D ( b i ) T ′′ . The quotient, denoted D ( b i ) T i , is isomorphic toSpec C [ c , . . . , b i , . . . , c m ], via c j = a j /b ji . The equation of E ( b i ) := E ∩ D ( b i ) T i is b i = 0. Over E ( b i ) ,the equation of T i is { y = Ψ( c , . . . , c i − , , c i +1 , . . . , c m )( x, z ) } ⊂ E ( b i ) × P (1 , , m ) . Henceforth, in our discussion we will identify the family T i → E ( b i ) with the affine spaceSpec[ c , . . . , ˆ c i , . . . , c m ]of in-homogeneous polynomialsΨ( c , . . . , c m )( x ) = x m + c x m − + · · · + c m of degree 2 m with c i = 1. Lemma 5.2. Given any point p ∈ E ( b ) , denote by D ( p ) the product of the deformation spaces ofthe singularities of the fiber ( T ) p . Then the family T → E ( b ) induces a smooth morphism from ananalytic neighborhood of p in E ( b ) to an analytic neighborhood of the origin in D ( p ) .Proof. Suppose p = ( c , . . . , c m ) ∈ ∆ { m , . . . , m r } . Set P := Ψ(1 , c , . . . , c m ) = Q ri =1 ( x − x i ) m i ,where x i satisfy P i x i m i = 0. The equation of ( T ) p is y = P ( x, z ) and so the singularities of ( T ) p are double points y = x m i . Hence, D ( p ) ∼ = C P ( m i − , with equisingular locus being 0 ∈ D ( p ).The first order deformations of p correspond to the first order deformation of P of the form P + ǫQ ,with Q being an arbitrary polynomial of degree 2 m − 3. Equisingular deformations are precisely thosesatisfying r Y i =1 ( x − x i ) m i − | Q ( x ) . Lemma follows from the fact that this divisibility condition imposes P ri =1 ( m i − 1) independent linearconditions on the coefficients of Q . (cid:3) The weighted projective space P (1 , , m ) is a projective cone over the rational normal curve of degree m . The linebundle O (1) is the restriction of the hyperplane section. Tangent cones.Lemma 5.3. Suppose W is a subvariety of T that is smooth at the origin, has dimension m , contains ∆ m , and whose tangent space is not contained in the hyperplane H m . Then the exceptional divisorof W in T ′′ is given by equations a m + i = b m + im + i = 0 , ≤ i ≤ m − . Here [ b : . . . : b m ] are homogeneous coordinates on E ⊂ T ′′ .Proof. The tangent space of W in T is a linear space of dimension m not contained in the hyperplane a m = 0, but containing { a m +1 = · · · = a m − = a m = 0 } . Therefore, the initial ideal of W satisfies in ( I ( W )) = ker[ m − I ∗ ] , where m − is the zero ( m − × ( m − 1) matrix, I is a row-permutation of the identity matrix I m − ,and ∗ is a column of complex numbers. Equivalently, in ( I ( W )) = ( a m + σ ( i ) + t i a m ) , ≤ i ≤ m − , t i ∈ C , σ ∈ S m − . Therefore, under the substitution a i = b ii , the initial ideal of π − ( W ) in T ′ becomes { b m + im + i = 0 , ≤ i ≤ m − } . (cid:3) Starting with the simple Lemma 5.3, we draw important corollaries regarding various geometricstrata inside T . Consider a point a ∈ ∆ { m , . . . , m n } withΨ a ( x ) = ( x − x ) m · · · ( x − x n ) m n where P ni =1 m i x i = 0. We set T ( i ) = Spec C [ c i,j ] ≤ j ≤ m i to be the versal deformation space of the m th i -order tacnode ( x, y ) = (0 , x i ) in the fiber Y a . Then we have a natural mapΦ : ( T, a ) → n Y i =1 T ( i ) , defined in a neighborhood of the point a ∈ T . Lemma 5.4. Consider the linear subspace W := { a m +1 = · · · = a m − = 0 } of T and a point a ∈ W . Then d Φ( T a W ) is not contained in the union of hyperplanes c i, m i = 0 . Remark 5.5. We can also reformulate the corollary as the statement that, for all i ,( T a W ) ∩ ( d Φ) − ( { c i, m i = 0 } ) = T a ∆ m . ⋄ Proof. The statement follows from the geometric interpretation of the hyperplane c i, m i = 0 as thetangent space to deformations vanishing at the point ( x, y ) = ( x i , a ( x ) = P ( x ),for some polynomial P ( x ). We note that the generic first order deformation ( P ( x ) + ǫQ ( x )) + ǫλ ofΨ a in W does not vanish at ( x i , 0) for all i . The proof is finished. (cid:3) Corollary 5.6. Consider W E := { b m +1 = · · · = b m − = 0 } ⊂ E . The following statements hold (1) W E ∩ (∆ m − ) E is analytically irreducible at every point of (∆ m ) E . (2) W E ∩ (∆ m − ) E is analytically irreducible at all points of strata ∆ { m , . . . , m n } E where ( m , . . . , m n ) is an arbitrary n -tuple with n ≥ .Proof. We first observe that both statements are equivalent to each of the analogous statements for T ′′ r E , T ′ r { } and T r { } , in turn.Working now on T r { } , we consider a point a ∈ ∆ { m , . . . , m n } and the induced mapΦ : ( T, a ) → n Y i =1 T ( i ) . m − = n [ i =1 Φ − ( n Y j =1 ∆ m j − δ i,j ) . By Lemma 5.4, we have T a Φ − ( Q nj =1 ∆ m j − δ i,j ) ∩ T a W = T a ∆ m . Therefore, ∆ m is, locally at a , asmooth component of the intersection Φ − ( Q nj =1 ∆ m j − δ i,j ) ∩ W . This establishes the first claim.The second claim is proved analogously. We observe that for n ≥ 3, the locus ∆ m − , locally at a , is a union of varieties, each mapping onto ∆ m i under Φ followed by the projection to some i th factor. (cid:3) Geometry of ∆ m − . We recall and reformulate Lemma 2.12 of [CH98b] in the following form: Lemma 5.7. Suppose λ is a non-zero number. For every positive integer m , there is a polynomial P m ( x ) = x m + α x m − + · · · + α m such that P m ( x ) − λ has m − double roots. Moreover, α k − = 0 for all k , α = 0 , and P m ( x ) is unique up to scaling α k ξ k α k , where ξ is an m th root of unity. We now reprove Lemma 4.1 of [CH98a]. Lemma 5.8. For any m -dimensional smooth variety W , containing ∆ m , whose tangent space is notcontained in the hyperplane a m = 0 , we have W ∩ ∆ m − = ∆ m ∪ Γ where Γ is a smooth curve tangent to ∆ m with order m at the origin.Proof. Recall that the morphism f : T → T is the weighted blow-up of T followed by the base changeof order 2. Consider the weighted projective tangent cones of W and ∆ m − , denoted respectively W E and (∆ m − ) E , inside the exceptional divisor E . Lemma 5.7 implies that away from the locus b = 0,the intersection of W E and (∆ m − ) E is a single point G , at least set-theoretically.From now on, we work on E ( b ) = Spec C [ c , . . . , c m ]. By Lemma 5.3, W E = { c m +1 = · · · = c m − = 0 } and hence T × E W E = { y = ( x m + x m − + c x m − + · · · + c m ) + c m ) } . Suppose coordinates of G in E ( b ) are ( λ , . . . , λ m ), where λ m = 0. We then haveΨ G = P ( x ) + λ m = Q ( x ) S ( x )where P ( x ) = x m + x m − + P mi =3 λ i x m − i , the polynomial Q ( x ) is monic of degree m − 1, withdistinct roots, and S ( x ) is a quadric. We will proceed now to show that intersection W E ∩ (∆ m − ) E is transverse at G .First, observe that the tangent space to ∆ m − at G is given by polynomials of degree 2 m − Q . The tangent space to W E at G consists of the first-order deformations( P ( x ) + ǫR ( x )) + λ m + ǫλ ′ m = Ψ G + ǫ (2 P ( x ) R ( x ) + λ ′ m ) , where R ( x ) is a polynomial of degree m − P ( x ) R ( x ) + λ ′ m is divisible by Q only if R = 0 and λ ′ m = 0. This is straightforward. Suppose 2 P ( x ) R ( x ) + λ ′ m isdivisible by Q . Then 4 P R + 4 λ ′ m P R + ( λ ′ m ) ≡ Q − λ m R + 4 λ ′ m P R + ( λ ′ m ) ≡ Q Observing that the left-hand side of the equality above has degree less that 2 m − Q ( x ),we conclude that − λ m R + 4 λ ′ m P R + ( λ ′ m ) = 0 . This implies that λ ′ m = 0 and R = 0.We proved that G = Γ ∩ E is a smooth point of the divisor E ⊂ T , Therefore Γ is a smooth curve,which in turn intersects the exceptional divisor E transversely at G .Let H := { a = 0 } and H m := { a m = 0 } be the coordinate hyperplanes in T . Then f ∗ H · Γ = ( H + 2 E ) · Γ = 2 E · Γ = 2 . H · Γ = 1.This necessarily implies that Γ is smooth near the origin in T . Similarly, equalities f ∗ H m · Γ = ( H + 2 mE ) · Γ = 2 mE · Γ = 2 m imply that H m · Γ = m . We finish by observing that ∆ m = W ∩ H m . (cid:3) It is enlightening to think of the family Z Γ := Z × T Γ as a stable reduction of the family Y Γ := Y × T Γ. The central fiber of Z Γ is the union of the normalization of y = x m and the( m − T ) G , while the generic fiber is an ( m − Z Γ is a family of generically smooth curves with the central fiber S T Figure 4. being the union of the normalization of y = x m and the rational curve attached at the points lyingover the tacnode of Y . From the construction, we can easily deduce that the total space of thenormalized family is smooth. It follows that any family of stable curves of genus g that globalizes( Z Γ ) ν intersects the boundary ∆ in M g with multiplicity 2 at the point G ∈ Γ. As Γ is a degree2 cover of Γ, ramified at G , we conclude that any family of stable curves of genus g that globalizes( Y Γ ) ν intersects the boundary of M g with multiplicity 1 at the origin. We state this more preciselyas Lemma 5.9. The central fiber of the family ( Y Γ ) ν → Γ has a single node (lying over the tacnode inthe central fiber of Y Γ ). Moreover, the total space of the family is smooth at this node. Geometry of ∆ m − . Let W be as above. Then we have W ∩ ∆ m − = ∆ m ∪ S , where S isresidual to ∆ m in the intersection. We would like to understand the geometry of S and the restrictionof the miniversal family to S . We do this by passing to T ′′ .If S E is the exceptional divisor of S , then we have W E ∩ (∆ m − ) E = (∆ m ) E ∪ S E . We study the geometry of the family T S := T × E S E of tails over S E . First of all, we have the followingresult that follows from Corollary 5.6: Corollary 5.10. S E ∩ (∆ m ) E = ⌊ m/ ⌋ [ i =1 ∆ { i, m − i ) } E . (5.11)From the above discussion, we know that a generic fiber of T S → S E is an ( m − m − |O (2) | on P (1 , , m ) and passing through points[ x : y : z ] = [1 : 0 : ± S νE tothe coarse moduli space M , . The map is defined, generically, by sending a point p ∈ ( S E ) ν to thenormalization of the tail ( T S ) p , marked at the points [1 : 0 : ± 1] of attachment to S . The first orderof business is to understand which fibers of T S are degenerate, i.e., have geometric genus 0 or less.0 Proposition 5.12. The only degenerate fibers of T S → S E are fibers over the points ∆ { i, m − i ) } E and (∆ m − ) E ∩ S E . Proof. The δ -invariant of the singularity y = x a is ⌊ a/ ⌋ . Therefore the geometric genus of a curvein a stratum ∆ { a , . . . , a k } E is m − − P i ⌊ a i / ⌋ .By Corollary 5.10, we have S E ∩ (∆ m ) E = S ∆ { i, m − i ) } E . On the complement of (∆ m ) E , wehave b m = 0. We next work on an open cover U := D (2 m ) T m ⊂ T m .Let S U = S E ∩ U . Then any fiber over S U has equation P ( x ) + 1, where P ( x ) = ( x m + c x m − + · · · + c m ) , (here c i = a i /b i m ) . Suppose P ( x ) + 1 has type ( m , . . . , m k ): P ( x ) + 1 = ( x − x ) m · · · ( x − x k ) m k . Differentiating, we obtain 2 P ( x ) P ′ ( x ) = ( x − x ) m − · · · ( x − x k ) m k − Q ( x ), for some polynomial Q ( X ). Hence P ′ ( x ) is a multiple of ( x − x ) m − · · · ( x − x k ) m k − . We conclude that m − ≥ X i m i − k = 2 m − k, or, equivalently, k ≥ m + 1.The degeneration in moduli occurs only when the geometric genus is 0 or less. Since P i m i = 2 m ,we can have X ⌊ m i ⌋ ≥ m − m , . . . , m n ). As we are interested in points outside∆ m , there should be precisely two odd numbers, and the only case in which this occurs, under theadditional restraint k ≥ m + 1, is when( m , . . . , m n ) = (2 , . . . , | {z } m − , , . This happens only at the point G = (∆ m − ) E ∩ S E , which is unique by the proof of Lemma 5.8. (cid:3) We would like now to understand the geometry of T S → S E around the points (∆ m − ) E ∩ S E and W E ∩ S E . We have the following Lemma 5.13. Let G = (∆ m − ) E ∩ S E . Then S E has m − smooth branches around G .Proof. We work on E ( b ) ⊂ T . The de-homogenization of the equation of T G is y = Ψ G ( x ) = P ( x ) + λ m = ( x − x ) · · · ( x − x m − ) S ( x ) , where P ( x ) = x m + x m − + P mi =3 λ i x m − i is a polynomial of degree m and S ( x ) is a monic quadric.The tangent cone to (∆ m − ) E at G is a union of linear spaces of polynomials vanishing at thesubset of m − x , . . . , x m − . These linear spaces correspond to deformations preservingall nodes except one. We conclude that (∆ m − ) E has m − G . To establishthe lemma, it remains to show that W E intersects all branches transversely.As we have seen in the proof of the Lemma 5.8, the first order deformations of P ( x ) + λ m in W E are of the form Ψ G ( x ) + ǫ (2 P ( x ) R ( x ) + λ ′ m )where R ( x ) is a polynomial of degree m − { x i , . . . , x i m − } of { x , . . . , x m − } , there is a unique, upto scaling, pair ( R ( x ) , λ ′ m ) such that( x − x i )( x − x i ) · · · ( x − x i m − ) (cid:12)(cid:12) P ( x ) R ( x ) + λ ′ m . (cid:3) T S → S E around a point p ∈ ∆ { i, m − i ) } E . Thesingularities of the fiber are tacnodes y = x i and y = x m − i ) . We denote T (0) := Def( y = x i )and T (1) := Def( y = x m − i ) ) to be the deformation spaces of these singularities. By Lemma 5.2,the family T S → S E induces an isomorphism of a neighborhood of p in E with a neighborhood ofthe origin in the product T (0) × T (1). Let H , i ⊂ T (0) and H , m − i ) ⊂ T (1) be the distinguishedhyperplanes (see Section 4.1 for the definition). By Lemma 5.4, locally at p , variety W E is identifiedwith a smooth m − V of T (0) × T (1), and by Lemma 5.4, the tangent spaceof V does not lie in the union of preimages of H , i and H , m − i ) .Therefore S E is identified with the curve Γ i,m − i residual to ∆ i × ∆ m − i in the intersection V ∩ (∆ i − × ∆ m − i − ) . This curve is an analog of a curve studied in the Lemma 5.8.Hence, to analyze the geometry of tails arising from a single tacnode, we are forced to considerthe analogous problems posed for the product of the deformation spaces of multiple tacnodes. We dothis in the next chapter. Note that to finish the analysis of S , we could simply use [CH98a, Lemma4.3] which describes the geometry of the curve Γ i,m − i . However, we will need a slightly more generalresult for the case of several tacnodes, and so we reprove Lemma 4.3 of [CH98a] in our Lemma 6.4.Finally, we describe the geometry of the family T S → S E in Section 6.2.6. Multiple tacnodes case In this chapter, we study the geometry of the product of the deformation spaces of n tacnodes. Let m = ( m , . . . , m n ) be a sequence of positive integer numbers. We define M := n Y i =1 m i and m = X ( m i − 1) + 1 . For 1 ≤ i ≤ n , let T ( i ) ∼ = C m i − be the versal deformation space of the tacnode y = x m i ,with coordinates ( a i, , . . . , a i, m i ). We set T = Q ni =1 T ( i ). Inside T , we denote the preimage of thehyperplane a i, m i = 0 by H i .In analogy with the case of a single tacnode, we first make a base change a i,j = b j Mmi i,j to arrive at the space T ′ = Spec C [ b i,j ].We let T ′′ = Bl T ′ , and denote by E ⊂ T ′′ the exceptional divisor of the blow-up. In analogy withthe single tacnode case, we also consider the quotients of T ′′ by groups˜ µ a,b = Y ( i,j ) =( a,b ) µ j Mmi ;we denote T ′′ // ˜ µ a,b by T a,b . Lemma 6.1. Consider any subvariety W ⊂ T , of dimension m = P ni =1 ( m i − 1) + 1 , that is smoothat the origin, contains ∆ m , and whose tangent space is not contained in the union of hyperplanes H i .Then the exceptional divisor W E of W in T ′′ is defined by equations a i,j = b j Mmi i,j = 0 , ≤ i ≤ n, m i + 1 ≤ j ≤ m i − a i, m i = λ i a , m , ≤ i ≤ n ;(6.3) where λ i are some non-zero complex numbers. Here, b i,j are homogeneous coordinates on E , and a i,j = b j Mmi i,j = 0 .Proof. The proof is a straightforward generalization of the proof of Lemma 5.3. (cid:3) Geometry of ∆ m − .Lemma 6.4. For any W ⊂ T , as in Lemma 6.1, we have W ∩ ∆ m − = ∆ m ∪ Γ m , where Γ m is a curve intersecting ∆ m with multiplicity M at the origin.Proof. First, we would like first to understand how W E and (∆ m − ) E intersect inside E . Any pointof intersection outside (∆ m ) E has homogeneous coordinates [ b i,j ] satisfying Equations (6.2)-(6.3).Moreover, by Lemma 5.7, b i, = 0. From now on, we work on the open affine E ( b , ) = Spec C [ c i,j ] ( i,j ) =(1 , ,c i,j := a i,j / ( b , ) j Mmi , inside the exceptional divisor of the map f : T , → T . Note that f is the weighted blow-up of T followed by the base change of order 2 m · · · m n .Set theoretically, W E ∩ (∆ m − ) E consists of points whose coordinates satisfy the following condi-tions:(1) c ij ’s satisfy Equations (6.2)-(6.3).(2) c , m = 0. This simply means that a point is not in ∆ m .(3) The polynomial ( x m + x m − + c , x m − · · · + c ,m ) + c , m has m − ≤ i ≤ n , the polynomial( x m + c i, x m − + · · · + c i,m i ) + c i, m i has m i − c ,j . Then Equation (6.3)determines all numbers c i, m i , for 2 ≤ i ≤ n . Finally, we again invoke Lemma 5.7 and condition (4)to conclude that each of the vectors ( c i, , c i, , . . . , c i, m i ) is determined up to the scaling by an m th i root of unity.Summarizing, W E ∩ (∆ m − ) E consists of m · · · m n points. The tangent cone argument, analogousto that of the single tacnode case, shows that the intersection is transverse at these points.We conclude that Γ m · E = m · · · m n . Observe that f ∗ ∆ m = ∆ m + 2 m m · · · m n E. An application of the projection formula givesdeg( f ) (Γ m · ∆ m ) = f (Γ m ) · ∆ m = Γ m · f ∗ ∆ m = Γ m · (∆ m + 2 m m · · · m n E )= 2 m m · · · m n (Γ m · E )= 2 m ( m · · · m n ) Since deg( f ) = 2 m · · · m n , we establish thatΓ m · ∆ m = n Y i =1 m i = M. (6.5) (cid:3) We observe that a subvariety W ⊂ T satisfying the conditions of Lemma 6.1, projects smoothly ontoa subvariety proj k ( W ) of T ( k ) satisfying the conditions of Lemma 5.8. It makes sense, therefore, to talkabout a curve Γ k inside T ( k ), defined as the residual to ∆ k,m k in the intersection proj k ( W ) ∩ ∆ k,m k − .We deduce the following Corollary 6.6. The projection proj k : Γ m → Γ k has degree Q i = k m i . Proof. Clearly, (proj k ) ∗ [Γ m ] = deg(proj k )[Γ k ]. Note that (proj k ) ∗ ∆ i,m i = ∆ m . Now using (6.5) andapplying the projection formula, we obtain the needed result. (cid:3) The geometry of the miniversal family. In this section, we continue the study of the geom-etry of the surface S d,δ ( α, β ) and the family Y from Chapter 4. We use the notations of the previoussection and Section 4.4 throughout. We consider the intersection of a subvariety W ⊂ T , satisfyingthe conditions of Lemma 6.1, with ∆ m − − e inside T . We have W ∩ ∆ m − − e = ∆ m ∪ S , where S is residual to ∆ m in the intersection. Generalizing the arguments of Chapter 5, we studythe geometry of S in what follows.Let Y ⊂ Spec C [ x, y ] × T ′′ be the pullback to T ′′ of the miniversal family over T (1). Recall that Y is given by the equation y = Ψ( a , , . . . , a , m )( x ) . Define the ideal sheaf I = (cid:18)(cid:16) I M/m E , x (cid:17) m , y (cid:19) . Set Z := Bl I Y and F : Z → T ′′ . As in the previous chapter, Bl I (Spec C [ x, y ] × T ′′ ) → T ′′ is a family of rational surfaces withfibers over T ′′ r E being affine planes, and fibers over the exceptional divisor E being the union of Bl ( x m ,y ) C and P (1 , Mm , M ) = Proj C [ x, y, z ]. Here, z stands for a local generator of I E , and C [ x, y, z ]is graded with deg x = Mm , deg y = M and deg z = 1.Consider a distinguished open affine of T ′′ : D ( b , ) T ′′ = Spec C [ c i,j ] , where c , = b , ,c i,j = b i,j b , , ( i, j ) = (1 , . The exceptional divisor on D ( b , ) T ′′ is given by b , = 0 and the restriction of Z to E is Z E := Z × T ′′ E = S ∪ T ;where S is { ( y/x m ) = 1 } ⊂ E × Bl ( x m ,y ) C and T is given by { y = Ψ (cid:18) , c M/m , , . . . , c M , m (cid:19) ( x, z M/m ) } ⊂ E × P (1 , M/m , M ) . We denote the strict transform of S in T ′′ by ˜ S and the exceptional divisor of ˜ S in T ′′ by S E . Lemma 6.7. We have S E ∩ (∆ m ) E = ⌊ m / ⌋ [ a =1 (∆ { a, m − a ) } × × · · · × E . Proof. This follows from the analogous statement for the single tacnode, proved in Lemma 5.10. (cid:3) Remark 6.8. It follows that S E is a curve in E . Hence S has pure dimension 2. ⋄ We now work on the quotient T , of T ′′ by ˜ µ (1 , . Set f : T , → T ′′ and Z , = Z // ˜ µ (1 , . Thedistinguished open affine D ( b , ) T , is isomorphic to Spec[ c i,j ], where c , = b , c i,j = a i,j ( b , ) j Mmi , ( i, j ) = (1 , . The exceptional divisor E ( b , ) is given by b , = 0, and so is isomorphic to Spec C [ c i,j ] ( i,j ) =(1 , .The quotient T , of T is given by equation y = Ψ(1 , c , , . . . , c , m )( x, z ) . P (1 , Mm , M ).Note that at any point p of (∆ { a, m − a ) } × × · · · × E , the fiber of the family T , → E ( b , ) has two tacnodes: of order a and m − a , respectively. We denote their deformation spaces by D (0) = Def( y = x a ) and D (1) = Def( y = x m − a ) ). By a slight abuse of notation, we also denote D ( i ) = Spec C [ c i,j ], for 2 ≤ i ≤ n , and think of D ( i ) as the deformation space of the order m i tacnode.By Lemma 5.2, the family T , induces an isomorphism between a neighborhood of the point p inSpec C [ c ,j ] ≤ j ≤ m and a neighborhood of the origin in D (0) × D (1). We conclude that at the point p ∈ (∆ { a, m − a ) } × × · · · × E , there is a local isomorphismΦ : E ( b , ) → n Y i =0 D ( i ) , such that the map from E ( b , ) to the product of the versal deformation spaces of the singularitiesof ( T , ) p is the composition of Φ and the projection to D (0) × D (1). By a slight reformulation ofLemma 5.4, the image Φ( W E ) satisfies the conditions of Lemma 6.1.We can now state the result that we will use in the next chapter to compute Discrep λS (∆ m ). Lemma 6.9. Under the isomorphism Φ , the exceptional divisor S E of S is identified with the curve Γ a,m − a,m ,...,m n , of Lemma 6.4, inside Q ni =0 D ( i ) . Observe that the family Z over the blow-up ˜ S the following property. Any m th -order tacnode,in any fiber, is a limit of m − Z → ˜ S has no points of indeterminacy along the preimage of ∆ m , at least after thenormalization of the base. By abuse of notation, denote the strict transform of S under f also by˜ S . It follows that the family Z , → ˜ S has no point of indeterminacy along the preimage of ∆ m .7. Calculation of the discrepancy We continue the discussion of the previous chapter. Recall that we are considering the geometry ofthe branch S of S d,δ ( α, β ) around the point of indeterminacy [ X ] ∈ S d,δ ( α, β ) ∩ H q , and the geometryof the family Y → S . The curve X has n “new” tacnodes y , . . . , y n . The branch S is defined as theclosure of the deformations of X such that y deforms to m − y i deforms to m i − i ≥ 2. We use the notations of Section 6.2 in what follows.In the neighborhood of the point y on X , the family Y is the pullback of the family Y → S under the isomorphism φ : S → S . Recall, that we have constructed a blow-up f : ˜ S → S , and afamily Z , → ˜ S with no points of indeterminacy along the preimage of ∆ m . We think of f as theresolution of the moduli map along ∆ m . To compute the discrepancy Discrep λS (∆ m ), we observe that f ∗ (∆ m ∩ S ) = ∆ m + 2 m m · · · m n S E . The coefficient of S E is 2 m m · · · m n because ∆ m is given by equation a i, m i = 0 (for any i ) and thepullback of a i, m i = 0 to T , vanishes to the order 2 m m · · · m n along E .Recalling that f is a map of degree 2 m · · · m n , we have Discrep λS (∆ m ) = (cid:18) (2 m m · · · m n ) S E · λ (cid:19) / deg f = m S E · λ. It remains to compute S E · λ . Recall that the family Z , over S E is a quotient of the union of twocomponents. One component, S , is an isotrivial family, and hence does not contribute to S E · λ . Theother component is the family of tails T , restricted to S E . Its quotient is T , .The family T , → S E is a family of generically hyperelliptic curves of arithmetic genus m − 1, withgenerically m − T ν := T , × S E ( S E ) ν → ( S E ) ν . It induces a regular map from ( S E ) ν to M , . By Proposition 5.12, the image curve intersects theboundary at the points of ( S E ) ν lying over S E ∩ (∆ m − ) E and S E ∩ (∆ m ) E . At the points of S E ∩ (∆ m − ) E , of which there are exactly m . . . m n by the proof of Lemma 6.4,the curve S E has m − M , transversely.5To calculate the multiplicity with which S E intersects the boundary at the points { p , . . . , p κ } of( S E ) ν lying over the point p ∈ (∆ { a, m − a ) } × × · · · × E , we need to understand the geometry of the family T ν around the tacnodes in the fibers. By Lemma 6.9, S E is identified, in a neighborhood of p , with the curve Γ a,m − a,m ,...,m n inside Q ni =0 D ( i ). Therefore,by Corollary 6.6, the curve ( S E ) ν maps with degree ( m − a ) m . . . m n onto the curve Γ a inside thedeformation space D (0) of the tacnode y = x a . Let r i , for 1 ≤ i ≤ κ , be the ramification index ofthe map ( S E ) ν → Γ a at the point p i . Then by Lemma 5.9, the surface T ν has an A r i − -singularity at the a th -order tacnodein the fiber ( T ν ) p i . Hence, this a th -order tacnode contributes r i to the intersection multiplicity of( S E ) ν with ∆ at the point p i . Remembering that κ X i =1 r i = ( m − a ) m . . . m n , and applying the same argument to the other tacnode, we conclude that the intersection number of( S E ) ν with the boundary at points lying over p is( m − a ) m · · · m n + am · · · m n = n Y i =1 m i . Noting that S E ∩ (∆ m ) E has exactly m − S E ∩ (∆ m − ) E and S E ∩ (∆ m ) E to the intersection number of ( S E ) ν with the boundary in M , :( S E ) ν · ∆ = ( m − m m · · · m n ) + ( m · · · m n )( m − m − m · · · m n ) . Therefore, Discrep λS (∆ m ) = m S E · λ = m (( S E ) ν · ∆) / 12= ( m − m m · · · m n ) / . Performing the same calculation for the surfaces S i , residual to ∆ m in the intersection ∆ m − − e i ∩ W, we arrive at the formula Discrep λS m (∆ m ) = n X i =1 Discrep λS i (∆ m ) = 112 n Y i =1 m i ! n X i =1 ( m i − ! . Together with Equations (4.6) and (3.13), this finishes the proof of Theorem 1.11.8. Examples and applications Slopes of effective divisor on M g . Define a slope of a curve C ⊂ M g by s ( C ) := C · ∆ C · λ . If C is a moving curve in M g , then the slope of C gives a lower bound on the slope s g of effectivedivisors on M g as defined in [HM90]. We have noted in the introduction that C d,δ irr is a moving curvein M g when 3 d − g − ≥ d − g − ≥ C d,δ irr with 3 d − g − − V d,δ (see [Har86]) and by work of Eisenbud and Harris (see [EH87]), we know that,in this case, deformations of C d,δ irr span an irreducible Brill-Noether divisor inside M g whose slope is6 + g +1 . Therefore, by a standard argument (cf. [HM90]), whenever 3 d − g − ≥ − 1, we have abound: s g ≥ min { g + 1 , s ( C d,δ irr ) } , here, as always, δ = (cid:0) d − (cid:1) − g .6Using the recursion of Theorem 1.11, we computed the slopes of curves C d,δ irr for d ≤ 16 with a helpof a computer. This gives us some lower bounds on s g for g ≤ Table 1. Slopes of C d,δ irr g d δ d − g − s ( C d,δ irr ) ≈ g d δ d − g − s ( C d,δ irr ) ≈ g = 2 and g = 3 are sharp. The bounds for g = 4 and g = 5 are betterthan those given in [HM90], but are still not sharp. We remark that there are examples of movingcurves in M g , providing sharp lower bounds, at least for small g . For g ≤ 6, see, for example, [CHS].Finally, even though we have nothing to say about the asymptotic behavior of the bounds producedby curves C d,δ irr , it would not be surprising if these bounds approached 0, as g approached ∞ .8.2. Codimension one numbers on V d,g . Consider the Severi variety V d,g of irreducible planecurves of degree d and geometric genus g . In [DH88], Diaz and Harris have computed a great numberof geometrically meaningful divisors on V d,g in terms of three standard classes (see loc. cit. for thenotations): A = π ∗ ( ω ) , B = π ∗ ( ω · D ) , C = π ∗ ( D ) , and boundary divisors ∆ and ∆ i,j . In particular, we recall the formulas for the classes of divisors ofcurves with cusps, triple points and tacnodes: CU = 3 A + 3 B + C − ∆ , (8.1) T N = (3( d − 3) + 2 g − A + ( d − B − C + 32 ∆ , (8.2) T R = (cid:18) d − d + 82 − g + 1 (cid:19) A − d − B + 23 C − 13 ∆ . (8.3)The number of curves in C d,δ irr with any codimension one behavior, which was studied in [DH88], isexpressed in terms of intersection numbers of C d,δ irr with A, B, C and ∆.We recall that C d,δ · A = N d,δ , and so can be computed by Caporaso’s and Harris’s recursion. Thenumber C d,δ · B can also be expressed in terms of degrees of Severi varieties (see [Vak98]). However,intersection numbers C d,δ · C were not known.By Mumford’s formula C = 12 λ − ∆and hence the intersection C d,δ · C is computed in terms of L d,δ = C d,δ · λ and intersections of C d,δ with boundary divisors.We remark that numbers B d,δ ( α, β ) := C d,δ ( α, β ) · B (8.4)can be computed using the following recursion.7 Proposition 8.5. The numbers B d,δ ( α, β ) satisfy the following recursion: B d,δ ( α, β ) = X k kB d,δ ( α + e k , β − e k )+ X (cid:18) I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ β (cid:19) B d − ,δ ′ ( α ′ , β ′ )+ 2 X k I β ′ − β (cid:18) αα ′ (cid:19)(cid:18) β ′ − e k β (cid:19) N d − ,δ ′ ( α ′ + e k , β ′ − e k ) (cid:19) , where the second sum is taken over all triples ( δ ′ , α ′ , β ′ ) satisfying | β ′ − β | + δ − δ ′ = d − .Proof. The formula follows from definitions and Theorem 1.6. (cid:3) Theorems 1.6 and 1.11, together with Proposition 8.5, allow us to compute inductively the inter-section numbers of C d,δ irr with all the standard divisor classes and, hence, to find solutions to a largeclass of codimension one enumerative problems on V d,δ irr . For example, one can compute the number ofirreducible plane curves of degree d with either a single cusp, a tacnode, or a triple point, and nodesas the only other singularities, passing through the appropriate number of general points in P .8.3. Enumerative applications.Example 8.6. A simple induction argument involving the recursion of Theorem 1.11 shows that L d, = 32 ( d − d − d − d + 1) ,L d, = 14 (9 d − d + 66 d + 333 d − d − d + 828) . We skip the proof of these formulas, and only note that it requires an introduction of auxiliaryfunctions, such as N d, (0 , ( d − , L d, (0 , ( d − , B d, = 3( d − d − d + 1) ,N d, = 12 (9 d − d + 12 d + 81 d − . Using Equation (8.1) and the equality ∆ · C d, = 2 N d, , we recover the number of cuspidal curves in C d, : CU = 3 N d, + 3 B d, + 12 L d, − N d, = 12( d − d − . The right-hand side is the classical formula for the degree of the cuspidal locus in P ( d ). ⋄ The numerical computations suggest that, for a fixed δ , the function L d,δ is a polynomial of degree2 δ + 2 in d , with the leading term δ δ ! . This should be compared with the G¨ottsche’s conjecture([G¨ot98]) that states that N d,δ is a polynomial of degree 2 δ in d . The conjecture was proved for δ ≤ d plane curves with a single triple point and δ nodes, where δ ≤ 3, passing throughan appropriate number of general points. The postulated number is a polynomial of degree 2( δ + 1) in d . Our methods allow us to compute the number of plane curves with a triple point and an arbitrarynumber of nodes. However, we cannot produce a closed form formula. We ran computations, with ahelp of a computer, for d ≤ 13 and δ ≤ 3, and our numbers agree with those of [KP99].In a different direction, for g = 0 , , , 3, the recursions for the codimension one “characteristic”numbers of plane curves were given by Vakil in [Vak98]. We note that our computations agree withthose of Vakil, presented in the table on page 19 of loc. cit.8 References [CH98a] Lucia Caporaso and Joe Harris. Counting plane curves of any genus. Invent. Math. , 131(2):345–392, 1998.[CH98b] Lucia Caporaso and Joe Harris. Parameter spaces for curves on surfaces and enumeration of rational curves. Compositio Mathematica , 113(2):155–208, 1998.[CHS] I. Coskun, J. Harris, and J. Starr. The effective cone of the Kontsevich moduli space. preprint, available at .[DH88] Steven Diaz and Joe Harris. Geometry of the Severi variety. Trans. Amer. Math. Soc. , 309(1):1–34, 1988.[EH87] David Eisenbud and Joe Harris. The Kodaira dimension of the moduli space of curves of genus ≥ Invent.Math. , 90(2):359–387, 1987.[G¨ot98] Lothar G¨ottsche. A conjectural generating function for numbers of curves on surfaces. Comm. Math. Phys. ,196(3):523–533, 1998.[Har86] Joe Harris. On the Severi problem. Invent. Math. , 84(3):445–461, 1986.[HM90] J. Harris and I. Morrison. Slopes of effective divisors on the moduli space of stable curves. Invent. Math. ,99(2):321–355, 1990.[HM98] Joe Harris and Ian Morrison. Moduli of curves , volume 187 of Graduate Texts in Mathematics . Springer-Verlag,New York, 1998.[KP99] Steven Kleiman and Ragni Piene. Enumerating singular curves on surfaces. In Algebraic geometry: Hirzebruch70 (Warsaw, 1998) , volume 241 of Contemp. Math. , pages 209–238. Amer. Math. Soc., Providence, RI, 1999.[KP04] Steven L. Kleiman and Ragni Piene. Node polynomials for families: methods and applications. Math. Nachr. ,271:69–90, 2004.[Vak98] Ravi Vakil. Enumerative geometry of plane curves of low genus, 1998. math.AG/9803007. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138 E-mail address : [email protected] URL ::