A support theorem for nested Hilbert schemes of planar curves
aa r X i v : . [ m a t h . AG ] F e b A SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANARCURVES
CAMILLA FELISETTI
Abstract.
Consider a family of integral complex locally planar curves. We show that under someassumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decompositiontheorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on therelative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We willshow that no summand is supported in positive codimension. Introduction
For the rest of this section curves are assumed to be complex, integral, complete and with locallyplanar singularities. We remind what locally planar singularities mean:
Definition 1.1.
Let C be a complex curve. We say that C has locally planar singularities if for every p ∈ C the completion ˆ O C,p of the local ring of C at p can be written asˆ O C,p = C [[ x, y ]] / ( f p )for some reduced series f p ∈ C [[ x, y ]].Let C be a curve of arithmetic genus p a ( C ) := H ( C, O C ).We consider the Hilbert scheme of points C [ m ] , which parametrizes length m finite subschemes of C .More precisely the m − th Hilbert scheme of points of C is defined as C [ m ] := { zero dimensional closed subschemes Z ⊂ C | dim( O C / I Z ) = m } where I Z is the ideal sheaf of Z . Hilbert schemes have been introduced by Grothendieck in [Gr] andare now the focus of several works in mathematics. For a general introduction to Hilbert schemes ofpoints and their properties we refer to [Ko, R]. In [AIK] and [BGS], these varieties are proved to benonsingular, complete, integral, m dimensional and locally complete intersections. Moreover there isa forgetful map ρ : C [ n ] → C ( n ) from the Hilbert scheme to the symmetric product of the curve thatmap any subscheme Z to his support. Such a map is an isomorphism of algebraic varieties when thecurve C is nonsingular, while it is birational for singular curves.We consider here the so called nested Hilbert scheme C [ m,m +1] of length m + 1 subschemes of C inwhich an ideal of colength 1 is fixed. More precisely we define C [ m,m +1] as C [ m,m +1] : = { ( z ′ , z ) | z ′ ∈ C [ m ] , z ∈ C [ m +1] , z ′ ⊂ z } = { ( I, J ) ideals of O C | I ⊂ J and dim( O C /J ) = m, dim( O C /I ) = m + 1 } Also, we can consider the relative versions of C [ m ] and C [ m,m +1] (see [Ko] for details), that is if π : C → B is a proper and flat family of curves we can define two families π [ m ] : C [ m ] → B, ( C [ m ] ) b = ( C b ) [ m ] π [ m,m +1] : C [ m,m +1] → B, ( C [ m,m +1] ) b = ( C b ) [ m,m +1] In [Sh], Shende proves that, under some assumptions on the basis, the total space of the relative Hilbertscheme C [ m ] is smooth. As a result, the decomposition theorem applied to the map π [ m ] asserts thatthe complexes Rπ [ m ] ∗ Q decomposes in the derived category of constructible sheaves D bc ( B ) as a directsum of shifted intersection complexes associated to local systems on constructible subsets of the base.Among them we find the intersection complex whose support is the whole base B . More precisely, ifwe denote by ˜ π : ˜ C → ˜ B the restriction of the family to the smooth locus, then any fiber is a smoothcurve and its Hilbert scheme coincides with the symmetric product; in particular the map ˜ π [ m ] issmooth. Hence the summand of Rπ [ m ] ∗ Q [ m + dim B ] with support equal to B is L IC B ( R i ˜ π [ m ] ∗ Q )[ − i ].Migliorini and Shende showed that this is in fact the only summand. Theorem ([MS1], Theorem 1) . Let
C → B be a proper and flat family of integral plane curves andlet ˜ π : ˜ C → ˜ B be its restriction to the smooth locus. If C [ m ] is smooth then Rπ [ m ] ∗ Q [ m + dimB ] = M IC B ( R i ˜ π [ m ] ∗ Q )[ − i ] . Here we prove that an analogous statement holds for the nested case.
Theorem 1.
Let
C → B be a proper and flat family of integral plane curves and let ˜ π : ˜ C → ˜ B be itsrestriction to the smooth locus. If C [ m,m +1] is smooth then Rπ [ m,m +1] ∗ Q [ m + 1 + dimB ] = M IC B ( R i ˜ π [ m,m +1] ∗ Q )[ − i ] . Versal deformations of curves singularities
As we will systematically employ versal deformation of curve singularities (as analytic spaces), werecall here some known results. For further details we refer to [GLS].
Definition 2.1.
Let (
X, x ) be the germ of a complex analytic space.(i) A deformation ( i, φ ) : (
X, x ) i −→ ( X , x ) φ −→ ( S, s ) is a morphism φ of germs of complex analyticspaces, together with an injection i such that X ∼ = i ( X ) = X x .(ii) A deformation ( i, φ ) : ( X, x ) i −→ ( X , x ) φ −→ ( S, s ) is called versal if, for a given deformation ( j, ψ )as above, the following holds: forn any closed embedding k : ( T ′ , t ) → ( T, t ) of complex germsand any morphism θ ′ : ( T ′ , t ) → ( S, s ) there exists a morphism θ : ( T, t ) → ( S, s ) satisfying(a) θ ◦ k = θ ′ , and(b) ( j, ψ ) = ( θ ∗ i, θ ∗ φ ).(iii) A deformation is locally versal if it induces versal deformations of all the singularities of X .(iv) A versal deformation is called miniversal if, with the notation of (iii), the Zariski tangent map T ( θ ) : T T,t → T ( S,s ) is uniquely determined by ( i, φ ) and ( j, ψ ).In the following section we will often use miniversal deformations since they can be described explic-itly. More precisely let ( C,
0) be the germ at the origin of the zero locus of some f ∈ C [ x, y ] such that f (0) = 0. Fix g . . . g t ∈ C [ x, y ] whose images form a basis of the vector space C [ x, y ] / ( f, ∂ x f, ∂ y f ).Then consider F : C t × C → C t × C given by F ( u , ..., u t , x, y ) = ( u , . . . , u t , ( f + g i u i )( x, y )). Takingthe fibre over C t × C t ; taking germs at the origin gives the miniversal de-formation ( C , → ( C t ,
0) of C . Moreover, if g ′ , . . . , g ′ s ∈ C [ x, y ] are any functions and ( C ′ , → ( C s , C , then the tangent map C s → C [ x, y ] / ( f, ∂ x f, ∂ y f ) is just in-duced by the quotient C [ x, y ] → C [ x, y ] / ( f, ∂ x f, ∂ y f ). As soon as this map is surjective, the family( C ′ , → ( C s ,
0) is itself versal.We would like to have a measure of ”how singular” a curve is, for example we could look at how far acurve is from its normalization. Given a singular curve C and denoted its normalization by C , we definethe cogenus δ to be the difference between its arithmetic and geometric genera δ ( C ) := p a ( C ) − p a ( C ).For example, the cogenus of a curve with one node is precisely 1. The following theorem, show whythe cogenus is a good candidate for our purpose. Moreover it will be the key result to reduce the proofof Theorem 1 to the case of a family of nodal curves. Theorem 2.1 ([T]) . Let
C → B be a family of curves. Then the cogenus is an upper semicontinuousfunction on B . Local versality is an open condition and in a locally versal family the locus of δ -nodalcurves is dense in the locus of curves with cogenus at least δ . In particular, the locus of curves ofcogenus δ in a locally versal family has codimension δ . As we are working with the cogenus we would like to have a result that allows us not to care about p a ( C ). In [L] Laumon showed that any curve singularity can be found on a rational curve. We will seethat there exist an analogous result for families, that is given a family of curves C → B then around apoint b ∈ B one can find a different family of rational curves such that C ′ b = C b and the two familiesinduce the same deformations of the singularities of the central fiber. This is a consequence of thefollowing proposition: Proposition 2.2 ([FGVs]) . The map from the base of a versal deformation of an integral locallyplanar curve to the product of the versal deformations of its singularities is a smooth surjection.
SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 3
Corollary 2.3 ([MS1], Cor. 6) . Let π : C → B be a family of curves. Fix b ∈ B , and let C b be thenormalization of C b .Then there exists a neighbourhood b ∈ U ⊆ B and a family π : C ′ → U such that C ′ b is rational with the same singularities as C b , and C and C ′ induce the same deformations of thesesingularities on U . In particular, they have the same discriminant locus. Moreover, on U, we have anequality of local systems R ˜ π ′∗ C ⊕ H ( C b ) , where H ( C b ) denotes the constant local system with thisfiber. To make use of such a replacement we need to know that C ′ [ m,m +1] is smooth if C [ m,m +1] is. Thisfollows from results on the smoothness of the nested Hilbert scheme which we are going to show. Theresults and their proof are closely analogous to [Sh, Prop. 17 and Thm.19], in which they are statedfor C [ m ] . 3. Smoothness of the relative nested Hilbert scheme
Let V ⊂ C [ x, y ] be a finite dimensional smooth family of polynomials and consider the family ofcurves C V := { ( f, p ) ∈ V × C | f ( p ) = 0 } . If we consider the associated family of nested Hilbert scheme C [ m,m +1] V then it is included in V × ( C ) [ m,m +1] . In [C], Cheah shows that the nested Hilbert scheme ( C ) [ m,m +1] is nonsingular for all m .Moreover she gives an explicit description of its tangent space: if ( I, J ) is a pair of ideals of C [ x, y ]with I ⊆ J such that ( I, J ) defines a point in ( C ) [ m,m +1] , then the tangent space T ( I,J ) ( C ) [ m,m +1] isisomorphic to Ker ( φ − ψ ) where φ : Hom C [ x,y ] ( I, C [ x, y ] /I ) → Hom C [ x,y ] ( I, C [ x, y ] /J ) ψ : Hom C [ x,y ] ( J, C [ x, y ] /J ) → Hom C [ x,y ] ( I, C [ x, y ] /J )are the obvious maps and( φ − ψ ) : Hom C [ x,y ] ( I, C [ x, y ] /I ) ⊕ Hom C [ x,y ] ( J, C [ x, y ] /J ) → Hom C [ x,y ] ( I, C [ x, y ] /J )is defined as ( φ − ψ )( η , η ) := φ ( η ) − ψ ( η ).Let us detail this isomorphism a little bit. The tangent space T J ( C ) [ m ] to the Hilbert scheme ( C ) [ m ] in an ideal J is canonically isomorphic to Hom C [ x,y ] ( J, C [ x, y ] /J ) and the isomorphism is constructedin the following way. Given an element η ∈ Hom C [ x,y ] ( J, C [ x, y ] /J ) we choose a lifting ˜ η : J → C [ x, y ]and such a lifting gives a tangent vector J ǫ,η = J + ˜ η ( J ). The fact that η is a morphism of C [ x, y ]-modules ensures that J ǫ,η is indeed an ideal of C [ x, y, ǫ ] / ( ǫ ) and thus that it defines a tangent vector.Now we observe that T ( I,J ) ( C ) [ m,m +1] ⊂ T I ( C ) [ m +1] ⊕ T J ( C ) [ m ] ∼ = Hom C [ x,y ] ( I, C [ x, y ] /I ) ⊕ Hom C [ x,y ] ( J, C [ x, y ] /J ) . The last isomorphism sends a pair ( η, ζ ) in a couple of tangent vectors( I ǫ,η , J ǫ,ζ ) with I ǫ,η = I + ˜ η ( I ) , J ǫ,ζ = J + ˜ ζ ( J ) , that do not satisfy the condition I ǫ,η ⊆ J ǫ,ζ a priori; this is ensured precisely by requiring that ( η, ζ )lies in Ker ( φ − ψ ).Choose a polynomial f ∈ I ⊂ J . If we write ( ˜ I, ˜ J ) for the image of the couple ( I, J ) in C [ x, y ] / ( f )then we have an exact sequence of vector spaces(1) 0 → T f, (˜ I, ˜ J ) C [ m,m +1] V → T f V × T ( I,J ) ( C ) [ m,m +1] → C [ x, y ] /I, where the last map is given by ( f + ǫg, ( η, ζ )) η ( f ) − g mod I. Even though ζ do not intervene explicitly in the last map, the condition η ( f ) − g ≡ I ensuresthat infinitesimally f + ǫg is contained in I ǫ,η . Since ( η, ζ ) ∈ Ker ( φ − ψ ), I ǫ,η ⊂ J ǫ,ζ ; thus f + ǫg belongs to J ǫ,ζ as well.Now, we observe that if f is reduced then all the fibers in a neighbourhood U of f are reduced andthe relative nested Hilbert schemes C [ m,m +1] U are reduced of pure dimension dim V + m + 1. Also theyare locally complete intersections [BGS]. Then C [ m,m +1] V is smooth at a point ( f, ( I, J )) if the tangentspace at this point has dimension m + 1 + dim V .Looking at dimensions of the vector spaces in (1), we notice that dim T f V = dim V as V is supposed CAMILLA FELISETTI to be smooth, dim T ( I,J ) ( C ) [ m,m +1] = 2 m + 2 by [C] and finally C [ x, y ] /I has dimension m + 1 byhypothesis: this tells us that dim T f, (˜ I, ˜ J ) C [ m,m +1] V = dim V + m + 1 if and only if the last map in (1) issurjective. The easiest way to ensure this is to ask for surjectivity already in the case η = ζ = 0, thatis T f V → C [ x, y ] /I is surjective.We are now ready to prove the smoothness of the relative nested Hilbert scheme. Proposition 3.1.
Let
C → V a family of versal deformations with base point ∈ V . For sufficientlysmall representatives C → V the relative nested Hilbert scheme C [ m,m +1] V is smooth.Proof. Suppose f is the polynomial defining C . Choose V ⊂ C [ x, y ] containing f such that C V → V isa versal deformation of the singularity of C and T f V contains all polynomials of degree ≤ m . Then T f V will be of dimension ≥ m + 1, thus for any I of colength m + 1, T f V will project surjectivelyonto C [ x, y ] /I . By the considerations above, the dimensions counting in (1) implies that the relativenested Hilbert scheme C [ m,m +1] V is smooth. (cid:3) Remark 1.
The smoothness of the relative nested Hilbert scheme over any versal deformation isequivalent to the smoothness over the miniversal deformations. In fact, if
C → V is the miniversaldeformations there are compatible isomorphisms V ∼ = V × ( C t ,
0) and
C ∼ = C × ( C t ,
0) and hence also C [ m,m +1] ∼ = C [ m,m +1] × ( C t , I, J ) with I of colength m + 1 , if we choose the basis V to be ( m + 1)-dimensional then the relative nested Hilbert scheme C [ m,m +1] V is smooth by proposition (3.1). Wewould like to find a basis We will need the following lemma, which is stated and proved in [Sh]. Lemma 3.2.
Let O be the completion of the local ring of a point on a reduced curve, and let O be afinite length quotient of O . Let W ⊂ O a generic k dimensional vector space. Then for I the imagein O of any ideal of colength ≤ k , we have W + I = O . With this lemma, we are now ready to prove the main theorem of this section.
Theorem 3.3.
Let ( C, be the analytic germ of a plane curve singularity and let ( C , → ( V , bean analytically versal deformation of ( C, . Then, for sufficiently small representatives C → V and ageneric disc ∈ D m ⊂ V , the space C [ h,h +1] D m +1 is smooth for h ≤ m + 1 .Proof. As in proposition (3.1) it is enough to prove the theorem for any versal deformation
C → V . Let( C,
0) be the analytic germ and let f ∈ C [ x, y ] be its equation. Choose g , . . . , g s ∈ C [ x, y ] such thattheir images in C [[ x, y ]] / ( f, ∂ x f, ∂ y f ) ∼ = C s form a basis. We have seen that the miniversal deformation C → V := C s has as fibres curves whose equation is of the form f + P t i g i = 0 . Let 0 ∈ D m +1 ⊂ V be a generic ( m + 1)-dimensional disc. Its tangent space W has dimension m + 1and lemma (3.2) ensures that W ⊂ C [[ x, y ]] / ( f, ∂ x f, ∂ y f ) is transverse to any ideal I of colength h ≤ m + 1. Thus for any h ≤ m + 1the final map of (1) is surjective, and C [ h,h +1] is smooth at pointsover 0 ∈ D m +1 which correspond to subschemes supported at the singularity.Finally let z ⊂ C [ h,h +1] be any subscheme of length h + 1 ; let z ′ be its component supported at thesingularity, say of length h ′ . Then an analytic neighbourhood of z in C [ h,h +1] differs from an analyticneighbourhood of z ′ in C [ h ′ ,h ′ +1] by a smooth factor. (cid:3) Corollary 3.4.
Let
C → B be a family of integral locally planar curves, locally versal at b ∈ B . Thenfor any generic, sufficiently small b ∈ D m +1 the relative nested Hilbert scheme C [ h,h +1] is smooth for h ≤ m .Proof. Such a situation is analytically locally smooth over that in theorem (3.3); a compactnessargument yields smoothness uniformly over an open neighbourhood in the base. (cid:3)
From the smoothness of the relative nested Hilbert scheme we can deduce an analogue result as theone in [MS1, Theorem 8].
Corollary 3.5.
Let
C → B a family of curves and let V be the product of the versal deformations ofcurve singularities. Then given a point b ∈ B ,(i) the smoothness of C [ m,m +1] depends only on the image T of T b B in T V ; (ii) if C [ m,m +1] is smooth along C [ m,m +1] b then dim T ≥ min ( δ ( C b ) , m + 1) ; SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 5 (iii) if dim
T ≥ m + 1 and T is general among such subspaces, then C [ m,m +1] is smooth C [ m,m +1] b ;(iv) C [ m,m +1] is smooth along C [ m,m +1] b for all m if and only if T is transverse to the image of theequigeneric ideal. It suffices for T to be generic of dimension at least δ ( C b ) .Proof. To prove ( i ) take a subscheme z ∈ C [ m,m +1] b which decomposes as z = ( z , . . . , z k )such that z ∈ C [ d ,d +1] b is a subscheme supported at a point c and z i ∈ C [ d i ] b are length d i subschemessupported on points c i .Let ( C i , c i ) → ( V i ,
0) be the miniversal deformations of the singularities ( C b , c i ) and ( B, b ) → Q ( V i , ` ( C i , c i ) → ( B, b ) pulls back. Then analytically locally, the germ ( C [ m,m +1] , [ z ])pulls back from ( C [ d ,d +1]0 , [ z ]) · Q ( C d i i , [ z i ]) along the same map. We observe that the fibres of( C d i i , [ z i ]) → ( V i ,
0) are reduced of dimension d i by [AIK] and the total space is nonsingular by [Sh,Prop. 17]. Moreover the same holds for ( C [ d ,d +1]0 , [ z ]) → V by proposition (3.1).As the V i were taken miniversal, the map T b B → Q T V i is uniquely defined and the smoothness ofthe pullback depends only on the image T of such a map. To check ( ii ) we might assume by ( i ) that themap T b B → Q T V i is an isomorphism and identify locally B with its image in some representatives B of Q ( V i , B until it can be written as B × D k for some polydisc D k ; as smoothnessis an open condition we may shrink D k further until C [ m,m +1] | B × ǫ is smooth for all ǫ ∈ D k . By [T], thelocus of nodal curves with the same cogenus as C b in Q V i is nonempty and of codimension δ ( C b );choose an ǫ such that B × ǫ contains the point p corresponding to such a curve. If m + 1 ≥ δ thestatement is trivial. If m + 1 ≤ δ , we can find a point z ∈ C [ m,m +1] p , which is a subscheme supported at m + 1 nodes. The Zariski tangent space T z C [ m,m +1] p has dimension 2 m + 2, therefore C [ m,m +1] p cannotbe smoothed over a base of dimension less than m + 1. For point ( iii ), we assume as above that B isembedded in B = Q V i . As the dimension of T is greater equal than m + 1, then by lemma (3.2) itis transverse to any ideal of colength ≤ m + 1, therefore the relative nested Hilbert scheme is smooth.Finally, ( iv ) if T in T V is transverse to the equigeneric ideal then the map in (1) is surjective for any I and the relative nested Hilbert scheme is smooth. (cid:3) Supports
For ease of the reader let us recall the statement of the decomposition theorem for nonsingularvarieties.
Theorem 4.1 ( Decomposition theorem ) . Let f : X → Y be a proper map of nonsingular complexalgebraic varieties. Then there exists a finite collection of constructible sets Y α and local systems L α on Y α such that the local system Rf ∗ Q [ dimX ] decomposes in the derived category of constructible sheavesas (2) Rf ∗ Q [ dimX ] ∼ = M α IC Y α ( L α )[dim X − dim Y α ] . Definition 4.1.
We call supports of f the Y α appearing in equation (2).We want describe the supports of the map π [ m,m +1] : C [ m,m +1] → B. Clearly among them we can always find the smooth locus ˜ B of the family and the summand supportedon B is given by the direct sum of the cohomology sheaves L IC B ( R i ˜ π [ m ] ∗ C )[ − i ], but a priori we couldhave other summands supported on subsets of positive codimension.In general, it is not easy to determine the supports of a given map f : X → Y . However, there exists afairly general approach to the so called support type theorems like the decomposition theorem, whichwas developed by Migliorini and Shende in [MS2]. Such an approach relies on the fact that eventhough a stratum S might be necessary in the stratification of a map f , the change in the cohomologyof the fibres of S can be predicted just by looking at the map on the strata containing S .Therefore, Migliorini and Shende constructed a coarser stratification, the stratification of higher dis-criminants . This description refines the notion of discriminant: instead of looking at the inverseimages of points one can consider the inverse images of discs D r of varying dimension r . Clearly CAMILLA FELISETTI the bigger the disc is the more likely its inverse image will be nonsingular. Let us be more precise:suppose Y is nonsingular and let Y = F Y α . Take y ∈ Y and let k be the dimension of the uniquestratum containing y . Consider the codimension k slice, meeting the stratum only in y . Its inverseimage will be a nonsingular codimension k subvariety of X . In case Y , we choose a local embedding( Y, y ) ⊂ ( C n ,
0) and we define a disc as the intersection of Y with a nonsingular germ of completeintersection T through y . The dimension of the disc is dim Y − codim T . Definition 4.2.
Keep the notation as above. We define the i − th higher discriminant ∆ i ( f ) as∆ i ( f ) := { y ∈ Y | there is no ( i − − dimensional disc φ : D i − → Y, with f − ( D i − ) non singular , and codim( D i − , Y ) = codim( f − ( D i − ) , X ) } The higher discriminants ∆ i ( f ) are closed algebraic subsets, and ∆ i +1 ( f ) ⊂ ∆ i ( f ) by the opennessof nonsingularity and the semicontinuity of the dimension of the fibres. Also we would like to remarkthat ∆ ( f ) is nothing but the discriminant ∆( f ) that is the locus of y ∈ Y such that f − ( y ) is singular.One advantage of higher discriminants is that they are usually much easier to determine via differentialmethod than the strata of a Whitney stratification. As we are supposing Y to be nonsingular, theimplicit function theorem prescribes precise conditions under which the inverse image of a subvarietyby a differentiable map is nonsingular: the tangent space of the subvariety must be transverse to theimage of the differential. Hence, under this assumption we have the following Proposition 4.2. ∆ i ( f ) := { y ∈ Y | for every linear subspace I ⊂ T y Y, with dim I = i − , the composition T x X df −→ T y Y → T y Y /I is not surjective for some x ∈ f − ( y ) } We may rephrase condition of proposition (4.2) saying that there is no ( i − I transverse to f .The following result shows the relevance of the theory of higher discriminants in determining thesummands appearing in the decomposition theorem. Theorem 4.3 ([MS2],Theorem B) . Let f : X → Y be a map of algebraic varieties. Then the set of i -codimensional supports of the map f is a subset of the set of i -codimensional irreducible componentsof ∆ i ( f ) . This theorem restricts significantly the set of candidates for the supports. Furthermore, to checkwhether a component of a discriminant is relevant it is enough to check its generic point.4.1.
Supports of π [ m,m +1] . We now want to construct a stratification of B such that the strata areprecisely the higher discriminants of the map π [ m,m +1] : C [ m,m +1] → B . Let b ∈ B be the base pointof B and suppose C b = C is the curve with the highest cogenus, which we call δ . For any i = 0 . . . δB i := { b ∈ B | δ ( C b ) = i } and we have that B = F i B i . As in the case of higher discriminants, we notice that B is thenonsingular locus of the family. We want to show the following proposition: Proposition 4.4.
Let π : C → B be proper flat family of curves such that the relative nested Hilbertscheme π [ m,m +1] : C [ m,m +1] → B is nonsingular for any m . Let δ be the highest cogenus we can findon a curve in the family. Then for any i = 0 . . . δ ∆ i ( π [ m,m +1] ) = B i . Proof.
Let b ∈ B i . As the relative nested Hilbert scheme is nonsingular at b , then by items ( ii ) − ( iv ) of corollary (3.5) then the image T of T b B into the product of the first order deformations ofthe singularities C b must be of dimension greater or equal than i . Therefore we have that B i ⊆ ∆ i ( π [ m,m +1] ). Conversely suppose b ∈ ∆ i ( π [ m,m +1] ). If the cogenus of C were < i , then T would havedimension < i contradicting item ( ii ) of corollary (3.5). (cid:3) As a consequence of theorem (4.3) if we have supports different from the smooth locus, then we willhave to look for them in the i -codimensional irreducible components of the B i ’s.We will prove Theorem 1 using the a criterion on supports coming from mixed Hodge theory: the stalks SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 7 of IC sheaves appearing in the decomposition theorem are endowed with a mixed Hodge structure;moreover Saito [Sa] proves that the isomorphism H k ( f − ( y )) = H k ( Rf ∗ Q ) y ∼ = M α H k ( IC Y α ( L α )) y in the decomposition theorem is actually an isomorphism of mixed Hodge structures. Whenever wehave a mixed Hodge structure H = ⊕ H i we can define the so called weight polynomial as w ( H )( t ) := X ( − i + j t i dim Gr Wi H j ∈ Z [ t ] . This polynomial has the additivity property, i.e. if Z ⊂ X is a closed algebraic subvariety of X then w ( H ∗ ( X ))( t ) = w ( H ∗ ( X \ Z ))( t ) + w ( H ∗ ( Z ))( t ) . We have the following criterion:
Proposition 4.5. [MS1, Prop. 15]
Suppose f : X → Y is a proper map between nonsingular algebraicvarieties. Let F be a summand of Rf ∗ Q [dim X ] . If, for all y ∈ Y we have that w ( F y [ − dim X ]) = w ( X y ) , then F = Rf ∗ Q [dim X ] . First we show the result for the Hilbert scheme in ([MS1]) with a direct computation, then weproceed to prove our theorem for the nested case. As we remarked above, the criterion can be verifiedjust on the generic points of the strata. By theorem (2.1) the generic points of the B i are the nodalcurves. Therefore we can reduce the proof of Theorem 1 to the case of a family of nodal curves. Usingcorollary 2.3 and the techniques in [MS1], one can suppose that all the curves are rational. As a resultthe geometric genus will coincide with the cogenus.5. Proof of theorem 1
Let π : C → B a proper flat family of rational nodal curves locally versal at a base point b ∈ B .Call δ := δ ( C b ). Consider the nodes { x , . . . , x δ } of the central fiber C b . Shrinking B if necessary, wecan assume the following facts:1) The discriminant locus is normal crossing divisor ∆ := S D i with i = 0 , . . . , δ , where D i is thelocus in which the i − th node x i is preserved.2) If b ∈ B is such that C b is nonsingular, then the vanishing cycles { α , . . . , α δ } associated with thenodes are disjoint.As the curve C b is irreducible, the cohomology classes in H ( C b ) of these vanishing cycles are linearlyindependent, and can then be completed to a symplectic basis { α , β , . . . , α δ β δ } . Let T i be thegenerators of the (abelian) local fundamental group π ( B \ ∆ , b ) ∼ = Z δ where T i corresponds to “goingaround D i ”. Then the monodromy defining the local system R ˜ π ∗ Q on B \ ∆ is given via the Picard-Lefschetz formula, and, in the symplectic basis above, the images of the generators of the fundamentalgroup in GL ( H ( C b )) = GL (2 δ, C ) are given by block diagonal matrices consisting of one Jordan blockof order 2 corresponding to a symplectic pair { α i , β i } and the identity elsewhere. Also, as the vanishingcycles are independent, we can consider R ˜ π ∗ Q as direct sum of δ modules V i of rank 2 whose basisis { α i , β i } . This makes much more easier to compute the invariants of any local system obtained bylinear algebra operations from R ˜ π ∗ Q . In our case we observe that, as C b is nonsingular then C [ m,m +1] b = C ( m,m +1) b = C ( m ) b × C b = C [ m ] b × C b . By the MacDonald formula for the cohomology of the symmetric product we have(3) R i ˜ π [ m ] ∗ Q = [ i ] M k =0 i − k ^ R ˜ π ∗ Q ( − k ) ∼ = R m − i ˜ π ∗ Q ( m − i )where ( − k ) denotes the weight shift of ( k, k ) in the mixed Hodge structure on the cohomology. Call thelinear algebra operation above S i,m . Applying the K¨unneth formula and recalling that the cohomologyof any curve C b in the smooth locus has a pure Hodge structure given by R ˜ π ∗ Q = Q R ˜ π ∗ Q ∼ = Q δ R ˜ π ∗ Q ∼ = Q ( − R i ˜ π [ m,m +1] ∗ Q ) b = (cid:16) ( R i ˜ π [ m ] Q ) ⊕ ( R i − ˜ π [ m ] ∗ Q ⊗ R ˜ π ∗ Q ) ⊕ ( R i − ˜ π [ m ] Q ( − (cid:17) b CAMILLA FELISETTI
Call T i,m the linear algebra operation we apply to on R ˜ π ∗ Q to obtain R ˜ π [ m,m +1] ∗ : T i,m ( R ˜ π ∗ Q ) := M j =0 S i + j,m ( R ˜ π ∗ Q ) ⊗ R j ˜ π ∗ Q Then there exists natural isomorphisms (cid:0) S i,m H ( C b ) (cid:1) π ( B \ ∆) ∼ = H (cid:16) IC B ( R i ˜ π [ m ] ∗ Q ) (cid:17) b (cid:0) T i,m H ( C b ) (cid:1) π ( B \ ∆) ∼ = H (cid:16) IC B ( R i ˜ π [ m,m +1] ∗ Q ) (cid:17) b between the monodromy invariants on S i,m H ( C b )( resp S i,m H ( C b ) ) and the stalk at b of the first non-vanishing cohomology sheaf of the intersection cohomology complex of R i ˜ π [ m ] ∗ Q (resp. R i ˜ π [ m,m +1] ∗ Q ).The decomposition theorem implies that H ∗ ( C [ m ] b ) and H ∗ ( C [ m,m +1] b ) contain respectively the Hodgestructures H m := M i (cid:0) S i,m H ( C b ) (cid:1) π ( B \ ∆) I m := M i (cid:0) T i,m H ( C b ) (cid:1) π ( B \ ∆) as a summand. We want to show that this is the unique summand by proving that the weightpolynomial of the cohomology of the nested Hilbert scheme of the C b is equal to the weight polynomialof H m . In that case the theorem will follow from proposition (4.5). Proposition 5.1.
Under the previous assumptions the following holds(i) w ( C [ m ] b ) = w ( H m ) (ii) w ( C [ m,m +1] b ) = w ( I m ) Remark 2.
Even though we are supposing for simplicity that the family of curves is locally versalaround b , we may weaken our hypotheses by just asking that the family is regular around b andthat the locus of nodal curves is dense.5.1. Hilbert scheme case.
Let π : C → B a locally versal deformation of a singular rational nodalcurve C b =: C . As a warm up for the nested case, we will compute the weight polynomial of C [ m ] andthe weight polynomial of the Hodge structure H m given by the monodromy invariants and show theyare equal thus proving theorem [MS1, Theorem 1].5.1.1. Computation of w ( C [ m ] ). To compute w ( C [ m ] ) we use power series to find a formula for theclass of C [ m ] in the Grothendieck group. First we notice that(5) X m q m h C [ m ] i = X m q m h C [ m ] reg i Y x i X q m h C [ m ] x i i As C reg = P \ δ regular points p , . . . p δ then X m q m h ( P ) [ m ] i = X m q m h C [ m ] reg i Y p i X q m h C [ m ] p i i Now observe that ( P ) [ m ] = P m ; also as the p i are regular points h C [ m ] p i i = 1 for all m and we have:1(1 − q )(1 − q L ) = X m q m h C [ m ] reg i − q ) δ ⇒ X m q m h C [ m ] reg i = (1 − q ) δ − (1 − q L )where L denotes the weight polynomial of the affine line.Now, in [R] Ran shows that C [ m ] x consists of m − P with m − Y x i X q m h C [ m ] x i i = (cid:16)X q m (( m − L + 1) (cid:17) δ = (cid:0) − q + q L (cid:1) δ (1 − q ) δ . SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 9
Substituting in equation (5), we get X m q m h C [ m ] i = (cid:0) − q + q L (cid:1) δ (1 − q )(1 − q L )The coefficient of q m in the series is given by(6) w ( C [ m ] ) = m X s =0 ( − s δ X t =0 (cid:18) δt (cid:19)(cid:18) ts − t (cid:19) L s − t · m − s X l =0 L l = m X s =0 ( − s δ X l =0 (cid:18) δt (cid:19)(cid:18) ts − t (cid:19) L s − t · L m − s +1 − L − . Computation of w ( H m ) . Let b a point in the smooth locus. We now need to compute theinvariants in the cohomology groups H i ( C b ) of the monodromy ρ : π ( B \ ∆) → H ( C b ). Also, werecall that all the vanishing cycles α i have weight 0, while β i have weight 2.Considering the MacDonald formula to compute the cohomology of Hilbert scheme, we just need tounderstand the invariants of V l H ( C b ) for any l ≥
0. As we observed before, H ( C b ) can be viewed as adirect sum of 2-dimensional representations V i on which a generator T j ∈ SL (2 δ, C ) of the monodromyacts as the identity if i = j and T i ( α i ) = α i , T i ( β i ) = α i + β i . Thus H ( C b ) = L δi =1 V i and we have(7) l ^ H ( C b ) = M l + ... + l δ = l l ^ V ⊗ . . . ⊗ l δ ^ V δ , ≤ l i ≤ . Also, as dim V i = 2 l i ^ V i = C if l i = 0 V i if l i = 1 C ( −
1) if l i = 2 . The only invariants of V i are the α i , of weight 0. In conclusion we have that for any i = 0 , . . . , m wehave I ( i, δ ) := w (cid:16) ( H i ( C [ m ] b )) π ( B \ ∆) (cid:17) = ( − i [ i ] X k =0 L k [ i − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − k − j (cid:19) L j where the index k is the one in MacDonald formula and j represents the number of second externalpower we take in (7).Summing over m and taking the duality in (3) into account we get w ( H m ) = m − X i =0 ( − i (1 + L m − i ) [ i ] X k =0 L k [ i − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − k − j (cid:19) L j (8) + ( − m [ m ] X k =0 L k [ m − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − jm − k − j (cid:19) L j (9) Proof of point (i) in Proposition 5.1.
We start looking at w ( H m ). First we notice that due to prop-erties of binomial coefficient, the sum over j goes to δ while the sum in k can go to infinity. Also wehave that (cid:0) δj (cid:1)(cid:0) δ − ji − k − j (cid:1) = (cid:0) δi − k − j (cid:1)(cid:0) i − k − jj (cid:1) .Setting l = i − k − j and applying the remarks above we get w ( H m ) = m − X i =0 ( − i (1 + L m − i ) ∞ X k =0 L k δ X l =0 (cid:18) δl (cid:19)(cid:18) li − k − l (cid:19) L i − k − l ++ ( − m ∞ X k =0 L k δ X l =0 (cid:18) δl (cid:19)(cid:18) lm − k − l (cid:19) L m − k − l Set s = i − k and split the sum in two parts with respect to the product with (1 + L m − i ). w ( H m ) = m X s =0 ( − s ∞ X k =0 L k δ X l =0 (cid:18) δl (cid:19)(cid:18) ls − l (cid:19) L s − l ++ m X s =0 ( − s L m − s ∞ X k =0 L − k δ X l =0 (cid:18) δl (cid:19)(cid:18) ls − l (cid:19) L s − l Taking out the sums in k and recalling that P ∞ k =0 L k = 11 − L w ( H m ) = 11 − L m X s =0 ( − s δ X l =0 (cid:18) δl (cid:19)(cid:18) ls − l (cid:19) L s − l + − L − L m X s =0 ( − s L m − s ∞ X k =0 L − k δ X l =0 (cid:18) δl (cid:19)(cid:18) ls − l (cid:19) L s − l == 11 − L m X s =0 ( − s (1 − L m − s +1 ) δ X l =0 (cid:18) δl (cid:19)(cid:18) ls − l (cid:19) L s − l which is precisely w ( C [ m ] ). (cid:3) Nested Hilbert scheme case.
As above suppose π : C → B is a locally versal deformation ofa singular rational nodal curve C b =: C . We now want to show point ( ii ) of proposition (5.1), toconclude the proof of theorem (1). Again, we compute the weight polynomials w ( C [ m,m +1] ), w ( I m )and show that their are equal.5.2.1. Computation of w ( C [ m,m +1] ) . We start by stratifying C [ m,m +1] b . As the weight polynomial de-pends only on the class in the Grothendieck group, we can work there. Let C ,reg := C \ { x , . . . , x δ } .We can consider the colength 1 ideal of C [ m,m +1] b as a copy of C [ m ] b to which we add a further point p ∈ C b [ m ] . Whenever we add a regular point p the class does not change, while when the point is anode we need to be careful about the number of occurrences of the node in the colength one ideal.In [R], Ran shows that the nested Hilbert scheme C [ k,k +1] x supported on one node, consists of 2 k − P with 2 k − h C [ k,k +1] x i = (2 k − L + 1.We stratify C [ m,m +1]0 with respect to the number of times the nodes appear in h C [ m ]0 i : h C [ m,m +1] b i = h C [ m ]0 × C b ,reg ) i + δ X i =1 m X k =0 h ( C − x i ) [ m − k ] × C [ k,k +1] x i i == h C [ m ] b × C b ,reg ) i + δ m X k =0 h ( C − x ) [ m − k ] × C [ k,k +1] x i We observe that for any k ≥ h C [ k,k +1] x i = h C [ k ] x i + k L . Making a substitution in theabove equation we get h C [ m,m +1] b i = h C [ m ] b × C b ,reg ) i + δ m X k =0 h ( C b − x ) [ m − k ] × C [ k ] x i + δ L m X k =0 k h ( C b − x ) [ m − k ] i . Since P mk =0 h ( C b − x ) [ m − k ] × C [ k ] x i = h C b [ m ] i , we have that the second term of the sum consistsprecisely of those δ copies of C b [ m ] which, added to the first term, give C b [ m ] × C b . Finally, we noticethat ( C b − × ) can be considered as a curve ˜ C with δ − p, q . Thenthe class of its Hilbert scheme can be computed as h ˜ C [ m ] i = P mk =0 (cid:2) ( C b − x ) [ m − k ] (cid:3) × C [ k ] p,q , where C [ k ] p,q is the Hilbert scheme with support p ∪ q . As p and q are regular points, h C [ k ] p,q i is just the number of SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 11 length non ordered k − ple in p, q , which is equal to k .In conclusion we can write(10) h C [ m,m +1] b i = C b [ m ] × C b + δ L h ˜ C [ m ] i Computation of w ( I m ). We remind that H i ( C [ m,m +1] b ) = H i ( C [ m ] b ) ⊕ H i − ( C [ m ] b ) ⊗ H ( C b ) ⊕ H i − ( C [ m ] b )( − . We notice that, by applying the MacDonald formula to second term we get H i − ( C [ m ] b ) ⊗ H ( C b ) = [ i − ] M k =0 i − − k ^ H ( C b ) ⊗ H ( C b )( − k )As a result we will have to find both the invariants of V l H ( C b ) and those of V l H ( C b ) ⊗ H ( C b ). Wehave seen how to find the invariants of V l H ( C b ) in the computation for the Hilbert scheme; whenlooking at the invariants of V l H ( C b ) ⊗ H ( C b ) we have to be more careful: there is more than justthe invariant of V l H ( C b ) times the invariant of H ( C b ).Let us be more precise: recall that H ( C b ) = L δi =1 V i and that we have(11) l ^ H ( C b ) = M l + ... + l δ = l l ^ V ⊗ . . . ⊗ l δ ^ V δ , ≤ l i ≤ . Also, as dim V i = 2 l i ^ V i = C if l i = 0 V i if l i = 1 C ( −
1) if l i = 2 . Thus(12) l ^ H ( C b ) ⊗ H ( C b ) = ( M l + ... + l δ = l l ^ V ⊗ . . . ⊗ l δ ^ V δ ) ⊗ ( V ⊕ . . . ⊕ V δ ) . By the considerations above, the monodromy invariants of summands of type V i ⊗ V j for i = j arejust an invariant of V i tensor an invariant of V j , while invariants of summands of type V V i ⊗ V j arejust the invariants of V i with shifted weight.The invariants which are not the tensor product of an invariant of V l H ( C b ) times an invariant of H ( C b ) come from the summands V i ⊗ V i = V V i ⊗ Sym ( V i ). These summands provide additionalinvariants of weight 2, which are those of V V i .As equation (12) is symmetric in the V i ’s it is sufficient to compute the invariants of( M l + ... + l δ = l l ^ V ⊗ . . . ⊗ l δ ^ V δ ) ⊗ V and multiply what we obtain by δ .If l = 1 then the formula we wrote for the Hilbert scheme still holds, while when l = 1 we have acertain number of invariants of weight 2 to take into account. w (cid:16) ( H i ( C [ m ] b ⊗ H ( C b ))) π ( B \ ∆) (cid:17) = δ [ i ] X k =0 L k [ i − k ] X j =0 (cid:18) δ − j − (cid:19)(cid:18) δ − ji − k − j (cid:19) L j ++ (1 + L ) (cid:18) δ − j (cid:19)(cid:18) δ − − ji − k − j − (cid:19) L j ++ (cid:18) δ − j (cid:19)(cid:18) δ − − ji − k − j (cid:19) L j The first term in the sum represents the case in which l = 2, the second one is the case of l = 1 andthe last one is l = 0. As in the previous formula, the index k is the one in the MacDonald formula, while the index j represents the number of l i = l that are equal to 2.Summing over i we get w ( I m ) = m X i =0 (1 + L m +1 − i ) [ i ] X k =0 L k [ i − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − k − j (cid:19) L j ++ δ [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δ − j − (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j + (1 + L ) (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j − (cid:19) L j ++ (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j (cid:19) L j + L [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j ++ ( − m +1 δ [ m ] X k =0 L k [ m − k ] X j =0 (cid:18) d − j − (cid:19)(cid:18) d − jm − k − j (cid:19) L j ++ (1 + L ) (cid:18) δ − j (cid:19)(cid:18) δ − − jm − k − j − (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − jm − k − j (cid:19) L j ++ 2 L [ m − ] X k =0 L k [ m − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − jm − − k − j (cid:19) L j . Looking at equation (10) we want to separate the invariants which are the tensor product of invari-ants of the Hilbert scheme and the invariants of the curve from those coming from the weight 2 partof the pieces V i ⊗ V i , which we will prove to be precisely the invariants of the Hilbert scheme of thecurves with δ − A = m X i =0 (1 + L m +1 − i ) [ i ] X k =0 L k [ i − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − k − j (cid:19) L j ++ δ [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δ − j − (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − ji − k − j − (cid:19) L j ++ (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j (cid:19) L j + L [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j ++ ( − m +1 L [ m − ] X k =0 L k [ m − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − jm − − k − j (cid:19) L j + δ [ m ] X k =0 L k [ m − k ] X j =0 (cid:18) δ − j − (cid:19)(cid:18) δ − jm − k − j (cid:19) L j ++ (cid:18) δ − j (cid:19)(cid:18) δ − − jm − − k − j (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j (cid:19) L j ++ 2 L [ m − ] X k =0 L k [ m − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − jm − − k − j (cid:19) L j . while the latter are B = δ L m X i =0 ( − i (1 + L m +1 − i ) [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δ − j (cid:19)(cid:18) δ − − ji − k − j − (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j (cid:19) L j +( − m +1 δ L [ m ] X k =0 L k [ m − k ] X j =0 + (cid:18) δ − j (cid:19)(cid:18) δ − − jm − k − j − (cid:19) L j . Lemma 5.2. A = h C b [ m ] × C b i = w ( H m )( L − δ + 1) SUPPORT THEOREM FOR NESTED HILBERT SCHEMES OF PLANAR CURVES 13
Proof.
First we notice that, due to properties of binomial coefficients, the quantity [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δ − j − (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − ji − k − j − (cid:19) L j + (cid:18) δ − j (cid:19)(cid:18) δ − − ji − − k − j (cid:19) L j is equal to [ i − ] X k =0 L k [ i − − k ] X j =0 (cid:18) δj (cid:19)(cid:18) δ − ji − − k − j (cid:19) L j = I ( i − , δ );thus A = m X i =0 ( − i (1 + L m +1 − i ) I ( i, δ ) + δI ( i − , δ ) + L I ( i − , δ ) + ( − m +1 (2 L I ( m − , δ ) + δI ( m, δ )) . Now, by setting t = i − m X i =0 ( − i (1 + L m +1 − i ) δI ( i − , δ ) + δ ( − m +1 I ( m, δ ) = − δ w ( H [ m ] ) . Also, m X i =0 ( − i (1 + L m +1 − i ) I ( i, δ ) + L I ( i − , δ ) + ( − m +1 L I ( m − , δ ) = m X i =0 ( − i (1 + L m +1 − i ) I ( i, δ ) + m X i =0 ( − i ( L + L m +2 − i ) + ( − m +1 L I ( m − , δ ) , setting t = i − L ) m − X i =0 ( − i L m − i ) I ( i, δ ) + m − X i =0 ( − i L m − i I ( i, δ )++ ( − m +1 L I ( m − , δ ) + ( − m I ( m, d )(1 + L ) + ( − m − (1 + L ) == (1 + L ) m − X i =0 ( − i L m − i ) I ( i, δ ) + ( − m I ( m, d ) ! = (1 + L ) w ( H m ) . (cid:3) Analogously, using properties of binomial coefficients and setting t = i − Lemma 5.3. B = δ L h ˜ C [ m ] i and this complete the proof of proposition (5.1) and Theorem 1. Acknowledgements.
I wish to thank my supervisor Luca Migliorini for suggesting me this problemas long as for the countless comments and corrections. Then I would like to thank Filippo Viviani andGabriele Mondello for the helpful comments and suggestions. Also I am grateful to Enrico Fatighenti,Giovanni Mongardi, Danilo Lewanski and Marco Trozzo for the support in writing this article.
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