Deformations of plane curves and Jacobian syzygies
aa r X i v : . [ m a t h . AG ] A ug DEFORMATIONS OF PLANE CURVES AND JACOBIANSYZYGIES
ALEXANDRU DIMCA AND GABRIEL STICLARU
Abstract.
We relate the equianalytic and the equisingular deformations of areduced complex plane curve to the Jacobian syzygies of its defining equation.Several examples and conjectures involving rational cuspidal curves are discussed. Introduction
Let C : f = 0 be a reduced plane curve in the complex projective plane P . Theequianalytic and the equisingular deformations of a plane curve singularity ( C, p ) arerather well understood, and the corresponding theory is briefly recalled in the secondsection following [8, 44]. On the other hand, the equianalytic and the equisingulardeformations of the plane curve C , encoded in the (possibly non-reduced) analyticsubspaces V ∗ d ( S , ..., S r ) of the projective space P N , are well understood when C isnodal, see [29, 37, 39], but much less in the general case, see [6, 24, 25, 26, 27, 28,30, 31, 37, 42, 43]. Here d is the degree of the curve C , S ,..., S r is the list of all the singularities of C , ∗ = ea or ∗ = es denotes the equianalytic or the equisingulardeformations, and N = d ( d + 3) /
2, such that P N parametrizes all the degree d plane curves. The main general results on these analytic subspaces V ∗ d ( S , ..., S r ) arerecalled below in Theorem 2.5.In the third section, we study the subspaces V ead ( S , ..., S r ) by restating Theorem2.5 in terms of the Jacobian syzygies of f and its Jacobian module N ( f ), following aresult by E. Sernesi, see [38, Corollary 2.2]. The main results here are Theorem 3.3,describing the tangent space T C V ead ( S , ..., S r ), and Theorem 3.8, giving a necessaryand sufficient condition for C to be unobstructed in terms of mdr ( f ), the minimaldegree of a Jacobian syzygy for f .In this paper we are particularly interested in the deformation theory of rationalcuspidal plane curves. As it is conjectured that any such curve is either free ornearly free, see Conjecture 2.7, we state a number of corollaries for free and nearlyfree curves, see for instance Corollary 3.6 and Corollary 3.10, and we discuss suchcurves in a number of examples, see for instance Example 3.7. In the fourth section,we recall first the classification of rational cuspidal plane curves with at least 3 cuspsinvolving three infinite families F Z ( d, a ), F Z ( k ) and F E ( k ) introduced by Flenner,Zaidenberg and Fenske in [21, 22, 23]. Concerning the curves in these three families, Mathematics Subject Classification.
Primary 14H50; Secondary 14B05, 14C05, 13D02,13D10.
Key words and phrases. equianalytic deformation, equisingular deformation, Jacobian ideal, eq-uisingular ideal, free curve, nearly free curve, rational cuspidal curve. we conjecture that all of them are free , and that the corresponding invariant mdr ( f )decides to which family the curve C : f = 0 belongs, see Conjecture 4.9 for details.This conjecture was verified for the first few curves in each of the three families, usingtheir description in terms of equations or parametrizations. Section 4 is completedby a brief discussion of the Rigidity Conjecture by Flenner and Zaidenberg, whichimplies in particular that a rational cuspidal plane curve C with at least 3 cusps isstrongly es -rigid, i.e. V esd ( S , ..., S r ) is smooth and coincide with the G -orbit of C ,where G = P GL (3 , C ) is the automorphism group of P , acting in the obvious wayon P N .In the final section we discuss the equisingular deformations of a reduced planecurve, the main result being Theorem 5.1. The subtle structure of the analyticsubspace V ∗ d ( S , ..., S r ) is highlighted in Examples 5.2 and 5.4, where the rationalunicuspidal curve C : f = y d + x d − z = 0 is deformed, for d = 5, and respectively d = 6. The corresponding subspace V ea ( S ) consists of two G -orbits, and it is smooth,while (the support of) V ea ( S ) consists of a single G -orbit, but V ea ( S ) is not reducedat all points. This latter fact is similar to classical examples of B. Segre of planecurves with many cusps, see for instance [42, Corollary 2.3] or [37, Example 4.7.10],and to Wahl’s example of a curve of degree d = 104 having 3636 nodes and 900 cusps,see [43] and [24, Examples 6.4 (6)], but surprisingly in our example the dimensionof V ea ( S ) is the expected dimension. The corresponding equisingular deformationssubspaces V es ( S ) and V es ( S ) are unions of the orbit G · C with families of G -orbitsparametrized by weighted projective spaces of dimension 1 and 2 respectively. Eachorbit in these families contains G · C in its closure, such that both analytic subspaces V es ( S ) and V es ( S ) turn out to be smooth of the expected dimension.The authors thank Gert-Martin Greuel, Karol Palka and Edoardo Sernesi for usefuldiscussions and remarks concerning this paper.2. Various prerequisites
Basic facts on deformations of isolated plane curve singularities.
Con-sider an isolated curve singularity ( C,
0) : g = 0 at the origin of C , defined by ananalytic function germ g ∈ O = C { u, v } . Let I ea ( C,
0) denote the ideal in O gen-erated by g and by the partial derivatives g u , g v of g with respect to u and v . Thenthe Tjurina algebra of g is the quotient T ( g ) = O /I ea ( C,
0) and its dimension, de-noted by τ ( g ) or τ ( C,
0) = τ ea ( C, g or of ( C, I ea ( C,
0) is called the equianalytic ideal of g . Moreover, it is known thatthe germ ( T ( g ) ,
0) is the base space of the miniversal deformation of the singularity( C,
0) : g = 0, see for details [8, 25, 26, 27, 44].There is a smooth germ ( ES ( g ) , ⊂ ( T ( g ) ,
0) corresponding to the equisingulardeformations of the singularity ( C, T ES ( g ) ⊂ T ( g ) = T T ( g ) is given by a quotient I es ( C, /I ea ( C, , where I es ( C,
0) is the equisingular ideal of g in O , see for details [8, 25, 26, 27, 44].The following example is taken from [44]. EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 3
Example 2.2.
Consider the weighted homogeneous singularity ( C,
0) : u p + v q = 0with 2 ≤ p ≤ q . We will denote this singularity by T p,q following [30]. Then theequianalytic ideal I ea ( C,
0) is generated by u p − and v q − , while the equisingular ideal I es ( C,
0) is generated by u p − , v q − and all the monomial u a v b with aq + bp ≥ pq . Thiscorresponds to the fact that a family of plane curve singularities over a reduced baseis equisingular if and only if it is µ -constant, and the Milnor number of an isolatedweighted homogeneous singularity is not changed by adding higher order terms. Remark 2.3.
Note that we have I es ( C,
0) = I ea ( C, C,
0) is a simple singularity, i.e. it is a singularity of type A n , D n , E , E or E in Arnold’s classification.One defines the equisingular Tjuriana number of the singularity ( C,
0) by theequality(2.1) τ es ( C,
0) = dim O /I es ( C, , and the equisingular modality of the singularity ( C,
0) by the equality(2.2) m es ( C,
0) = dim I es ( C, /I ea ( C,
0) = τ ( C, − τ es ( C, . The modality mod ( g ) of the germ g with respect the the right-equivalence is alsorelated to these invariants by the formula(2.3) mod ( f ) = µ ( C, − τ es ( C, , see [25, Lemma 1.7].2.4. Basic facts on deformations of reduced plane curves.
Let S = C [ x, y, z ]be the polynomial ring in three variables x, y, z with complex coefficients and let P ( S d ) be the projective space parametrizing the degree d curves in P . Let N =dim P ( S d ) = d ( d + 3) /
2. Fix a list S , S , ..., S r of isomorphism classes of isolatedplane curve singularities. We consider two settings, namely the equianalytic set-ting, denoted with ea , and the equisingular setting, denoted with es . We denoteby V ∗ d ( S , ..., S r ) the subset in P ( S d ) corresponding to the reduced curves C ⊂ P ,having precisely r singularities which are required to be(1) isomorphic to the singularities S , ..., S r when ∗ = ea , and(2) equisingular to the singularities S , ..., S r when ∗ = es .By definition we have(2.4) V ead ( S , ..., S r ) ⊂ V esd ( S , ..., S r ) , with equality when all the singularities S j ’s are simple.Each of the two sets V ∗ d ( S , ..., S r ) has in fact the structure of a natural (possiblynon reduced) complex analytic subspace V ∗ d ( S , ..., S r ) of the projective space P N ,such that the restriction of the universal family over P N to the analytic subspace V ∗ d ( S , ..., S r ) has the expected functorial properties. The first rigorous proof of theexistence of this analytic subspace structure was given by J. Wahl in [43], for curves ALEXANDRU DIMCA AND GABRIEL STICLARU having only nodes A and simple cusps A as singularities. The general constructionfor the analytic subspace structure of V ead ( S , ..., S r ) was settled by G.-M. Greuel andU. Karras in [24], see Theorem 1.3 and Theorem 2.2. This proof was then adaptedfor the analytic subspace structure of V esd ( S , ..., S r ) by G.-M. Greuel and C. Lossenin [25]. Another, more direct approach to the existence of the analytic subspacestructure of V ∗ d ( S , ..., S r ) is given in the forthcoming book [28], see Theorems II. 2.32and II.2.36, where the authors also show that the reduction of these subspaces arequasi-projective subsets of P N . It is very likely that the analytic subspace structures V ∗ d ( S , ..., S r ) themselves are (algebraic) subschemes of P N , but this fact is provedonly in the case of curves with nodes and cusps as singularities, see [43, Theorem3.3.5].Consider now a reduced curve C : f = 0 in P , having r singular points located atthe points p j ∈ P , for j = 1 , ..., r . To such a curve we associate two 0-dimensionalsubschemes of P , namely Z ∗ ( C ) for ∗ = ea, es , such that the support of the schemes Z ∗ ( C ) is the singular points of C and one has by definition O Z ∗ ( C ) ,p j = O P ,p j /I ∗ ( C, p j ) . By our discussion in the previous subsection, it follows thatdeg Z ∗ ( C ) = X j =1 ,r τ ∗ ( C, p j ) . Let I ( Z ∗ ( C )) be the ideal sheaf in O P which defines the scheme Z ∗ ( C ). Note that Z ∗ ( C ) can also be regarded as a subscheme of the curve C , and then it is defined bythe ideal sheaf I ( Z ∗ ( C ) | C ) = I ( Z ∗ ( C )) ⊗ O C ⊂ O C . As noted in [26, Remark 2.4], one has an exact sequence0 → I C ( d ) → I ( Z ∗ ( C ))( d ) → I ( Z ∗ ( C ) | C )( d ) → , where I C ⊂ O P denotes the sheaf ideal of C , and one has O P ≃ I C ( d ). It followsthat H ( I ( Z ∗ ( C ) | C )( d )) = H ( I ( Z ∗ ( C ))( d )) /H ( O P ) . These sheaf ideals are important in view of the following result.
Theorem 2.5.
Let C be a reduced plane curves such that C ∈ V ∗ d ( S , ..., S r ) , with ∗ = ea, es . Then the following hold. (1) H ( I ( Z ∗ ( C ) | C )( d )) is isomorphic to the Zariski tangent space of the analyticsubspace V ∗ d ( S , ..., S r ) at the point C . (2) H ( I ( Z ∗ ( C ) | C )( d )) = 0 if and only if the analytic subspace V ∗ d ( S , ..., S r ) issmooth at the point C , of the expected dimension, namely N − deg Z ∗ ( C ) = d ( d + 3) / − X j =1 ,r τ ∗ ( C, p j ) . The claim (1) above is not very difficult and can be found in [8, Proposition 4.19].The second claim (2) is more difficult, and we refer to [26] and to the forthcomingbook [28], see Theorems II. 2.38 and II.2.40 for the complete proofs.
EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 5
Jacobian ideal, Jacobian module, and free and nearly free curves.
We denote by J f the Jacobian ideal of f ∈ S d , i.e. the homogeneous ideal in S spanned by the partial derivatives f x , f y , f z , and by M ( f ) = S/J f the correspondinggraded quotient ring, called the Jacobian (or Milnor) algebra of f . Let I f denote thesaturation of the ideal J f with respect to the maximal ideal m = ( x, y, z ) in S andconsider the local cohomology group, usually called the Jacobian module of f , N ( f ) = I f /J f = H m ( M ( f )) . The graded S -module AR ( f ) = AR ( C ) ⊂ S of all Jacobian relations of f is definedby(2.5) AR ( f ) k := { ( a, b, c ) ∈ S k | af x + bf y + cf z = 0 } . Its sheafification E C := ^ AR ( f ) is a rank two vector bundle on P , see [13, 38] for de-tails. More precisely, one has E C = T h C i ( − T h C i is the sheaf of logarithmicvector fields along C as considered for instance in [13, 38]. We set ar ( f ) m = dim AR ( f ) m = dim H ( P , E C ( m )) and n ( f ) m = dim N ( f ) m for any integer m . The minimal degree of a Jacobian relation for the polynomial f isthe integer mdr ( f ) defined to be the smallest integer m ≥ ar ( f ) m > mdr ( f ) = 0 if and only if, up-to a linear change of coordinates in G , f is independent of z , i.e. the curve C consists of d concurrent lines. We assume from now on in this paper that mdr ( f ) > C as well as the line arrangements with mdr ( f ) = 1are classified, see [20]. Hence the interesting cases are the curves with ar ( f ) = 0.It was shown in [12, Corollary 4.3] that the graded S -module N ( f ) satisfies aLefschetz type property with respect to multiplication by generic linear forms. Thisimplies in particular the inequalities0 ≤ n ( f ) ≤ n ( f ) ≤ ... ≤ n ( f ) [ T/ ≥ n ( f ) [ T/ ≥ ... ≥ n ( f ) T ≥ , where T = 3 d −
6. We set as in [10] ν ( C ) = max j { n ( f ) j } , and introduce a new invariant for C , namely σ ( C ) = min { j : n ( f ) j = 0 } . The self duality of the graded S -module N ( f ), see [38, 41], implies that n ( f ) s = 0exactly for s = σ ( C ) , ..., T − σ ( C ).Recall that C is a free curve if J f = I f , or equivalently ν ( C ) = 0, see [9, 16, 40].Note that for a free curve C : f = 0 of degree d , one has the exponents ( d , d ) where d = mdr ( f ) and d = d − − d , as well as the formula(2.6) τ ( C ) = ( d − − d d . Similarly, C is a nearly free curve if ν ( C ) = 1, see [1, 9, 17]. Note that one has σ ( C ) = d + mdr ( f ) − ALEXANDRU DIMCA AND GABRIEL STICLARU for a nearly free curve C : f = 0 of degree d , one has the exponents ( d , d ) where d = mdr ( f ) and d = d − d , and the formula(2.7) τ ( C ) = ( d − − d ( d − − . Our interest in the free and nearly free curves comes from the following.
Conjecture 2.7.
A reduced plane curve C : f = 0 which is rational cuspidal iseither free, or nearly free.This conjecture is known to hold when the degree d of C is even, or when d ≤ Jacobian syzygies, Jacobian module and equianalytic deformations
The following result is obvious, but plays a key role in the sequal.
Lemma 3.1.
The ideal sheaf J f associated to the homogeneous ideal J f , coincideswith the ideal sheaf I f associated to the homogeneous ideal I f , and with the idealsheaf I ( Z ea ( C )) . This ideal sheaf defines the scheme structure of the singular locusof the plane curve C . Denote by J f | C = J f ⊗ O C the resctriction of the ideal sheaf J f to the curve C and note that in view of Lemma 3.1 we have J f | C = I ( Z ea ( C ) | C ) . Hence, Theorem 2.5 can be restated in the case ∗ = ea as follows. Theorem 3.2.
Let C be a reduced plane curves such that C ∈ V ead ( S , ..., S r ) . Thenthe following hold. (1) H (( J f | C )( d )) is isomorphic to T C V ead ( S , ..., S r ) , the Zariski tangent spaceof the analytic subspace V ead ( S , ..., S r ) at the point C . (2) H ( J f | C ( d )) = 0 if and only if the analytic subspace V ead ( S , ..., S r ) is smoothat the point C , of the expected dimension d ( d + 3) / − τ ( C ) , where τ ( C ) = P j =1 ,r τ ( C, p j ) is the total Tjurina number of the curve C . In this section we look at the dimensions h (( J f | C )( d )) = dim H (( J f | C )( d )) and h ( J f | C ( d )) = dim H ( J f | C ( d )), using results by Sernesi in [38] and our study ofthe Jacobian syzygies and Jacobian module N ( f ) in [13]. EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 7
We recall the following diagram from [38], with exact rows and columns.(3.1) 0 00 / / T h C i / / T P O O η / / ( J f | C )( d ) O O / / / / E C (1) ∼ = O O / / O P (1) O O ∂f / / J f ( d ) / / O O O P O O f / / I C ( d ) O O / / O O O O Here the morphism ∂f is induced by the morphism of graded S -modules S (1) → J f ( d ) , ( a, b, c ) af x + bf y + cf z , and η is induced by ∂f . The middle column is the usual free resolution of the tangentbundle T P to P , and implies h ( T P ) = 3 h ( O P (1)) − h ( O P ) = 3 · − P GL (3 , C ) . The first exact row yields the following exact sequence0 → AR ( f ) → H ( T P ) → H ( J f | C )( d )) → H ( T h C i ) → . Indeed, one has H ( T h C i ) = H ( E C (1)) = AR ( f ) and H ( T P ) = 0 using themiddle column in the above diagram. Note that one has the following facts.(1) The automorphism group of P is G = P GL (3 , C ), and this group acts inan obvious way on the projective space P ( S d ), such that all the subsets V ∗ d ( S , ..., S r ) are unions of G -orbits. In particular, one has the inclusions T C ( G · C ) ⊂ T C V ead ( S , ..., S r ) ⊂ T C V ead ( S , ..., S r ) . (2) The tangent space T e G to G at the identity element e ∈ G can be identifiedto H ( T P );(3) Let H ⊂ G be the subgroup of elements fixing the polynomial f ∈ P ( S d ).Then the tangent space T e H to H at the identity element e ∈ G can beidentified to H ( T h C i ) = H ( E C (1)) = AR ( f ) , see for instance [20].(4) Hence the cokernel of the inclusion AR ( f ) → H ( T P ) can be identified tothe tangent space T C ( G · C ) to the orbit of C , at the point C .It follows from [38, Proposition 2.1] that one has H ( T h C i ) = N ( f ) d . Therefore weget the following result, which is our reformulation of [38, Corollary 2.2]. Theorem 3.3.
Let C be a reduced plane curves such that C ∈ V ead ( S , ..., S r ) . Thenthe Zariski tangent space T C V ead ( S , ..., S r ) of the analytic subspace V ead ( S , ..., S r ) atthe point C sits in the following exact sequence → T C ( G · C ) → T C V ead ( S , ..., S r ) → N ( f ) d → . ALEXANDRU DIMCA AND GABRIEL STICLARU
In particular, one has dim T C V ead ( S , ..., S r ) = 8 − ar ( f ) + n ( f ) d . Definition 3.4.
We say that the reduced plane curve C is ∗ -projectively rigid if T C ( G · C ) = T C V ∗ d ( S , ..., S r ). Note that this is equivalent to the equality of ana-lytic space germs ( G · C, C ) = ( V ∗ d ( S , ..., S r ) , C ) = ( V ∗ d ( S , ..., S r ) , C ) . We say thatthe reduced plane curve C is strongly ∗ -projectively rigid, if the analytic subspace V ∗ d ( S , ..., S r ) containing C is smooth and coincides with the orbit G · C .In particular, if C is ∗ -projectively rigid, it follows that the analytic subspace V ∗ d ( S , ..., S r ) is reduced and smooth at C . Clearly, if a curve C is (strongly) es -rigid,then C is also (strongly) ea -rigid. We have also the following result, with an obviousproof. Lemma 3.5.
A reduced plane curve C is (strongly) ∗ -projectively rigid if and only ifany curve C ′ in the G -orbit G · C is (strongly) ∗ -projectively rigid. If the reduced planecurve C ∈ V ∗ d ( S , ..., S r ) is ∗ -projectively rigid, then the G -orbit G · C is an irreducibleand connected component of the analytic subspace V ∗ d ( S , ..., S r ) , consisting only ofsmooth points of this analytic subspace. Corollary 3.6.
A reduced curve C : f = 0 is ea -rigid if and only if n ( f ) d = 0 . Inparticular, one has the following. (1) Any free curve is ea -rigid. (2) A nearly free curve C : f = 0 is ea -rigid if and only if one of the followingcases occurs:(i) mdr ( f ) ≥ , or(ii) mdr ( f ) = 2 and d = 4 , or(iii) mdr ( f ) = 1 and d = 2 , . The claim (2) above corrects the final claim in [17, Corollary 2.17].
Proof.
To prove the second claim, note that n ( f ) d = 0 if either d < σ ( C ) = d + mdr ( f ) − d > T − σ ( C ). The first inequality is equivalent to mdr ( f ) ≥
4. Thesecond inequality is equivalent to mdr ( f ) > d −
3. For a nearly free curve one has mdr ( f ) ≤ d/
2, see [17], and this gives the subcases (ii) and (iii) above. The followingexamples shows that the sub cases (ii) and (iii) above really occur. (cid:3)
Example 3.7.
In this example we list the free and nearly free curves C : f = 0 ofdegree d such that 2 ≤ d ≤ mdr ( f ) >
0. For details we refer to [30, 32]. Allof them are strongly ea -rigid except for the curve C ′ discussed below.(1) For d = 2, the smooth conic C : f = x + y + z = 0 is nearly free, with ar ( f ) = 3 and mdr ( f ) = 1. The scheme V ∗ ( ∅ ) coincides with the orbit G · C , and it is an Zariski open subset in P = P ( S ). Moreover, n ( f ) k = 0for k = 0, so the formula for dim V ∗ ( ∅ ) in Theorem 3.3 holds.(2) For d = 3, the cuspidal cubic C : f = x y + z = 0 is nearly free, with ar ( f ) = 1 and mdr ( f ) = 1. The quasi-projective variety V ∗ ( A ) coincideswith the orbit G · C , and it has codimension 2 in P = P ( S ). Moreover, n ( f ) = 0, so the formula for dim V ∗ ( A ) in Theorem 3.3 shows that the EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 9 analytic subspace V ∗ ( A ) is smooth. The triangle C ′ : f = xyz = 0 is free,with ar ( f ) = 2 and mdr ( f ) = 1. The analytic subspace V ∗ (3 A ) is smooth,coincides with the orbit G · C ′ , and has dimension 6.(3) For d = 4, there are 4 lists of singularities which may occur on a nearly freeirreducible quartic, namely 3 A , A A , A and E . The analytic subspace V ∗ (3 A ), V ∗ ( A A ) and V ∗ ( A ) are smooth and coincide with the correspond-ing G -orbits, while V ∗ ( E ) is still smooth, but the union of two G -orbits.Consider the two quartics C : f = y − xz − y z = 0 and C ′ : f ′ = y − xz = 0 . For all these quartics, except for C ′ , one has mdr ( f ) = 2, which implies ar ( f ) = n ( f ) = 0, and hence all these G -orbits are 8-dimensional in P = P ( S ). For C ′ one has ar ( f ) = n ( f ) = 1, and hence C ′ is not ea -rigid, sincedim T C ′ ( G · C ′ ) = 7 < T C ′ V ∗ ( E ) . Note also that the orbit G · C ′ is contained in the closure of the orbit G · C . It follows that the variety V ∗ ( E ) is the union of these two orbits, andhence the corresponding analytic subspace V ∗ ( E ) is smooth and irreducibleof dimension 8 at any point.The line arrangement C ′ : f = xyz ( x + y + z ) = 0 is also nearly free, theorbit G · C ′ is 8-dimensional as above, and coincides with V ∗ (6 A ). The linearrangement C ′′ : f = xyz ( x + y ) = 0 is free with mdr ( f ) = ar ( f ) = 1,the orbit G · C ′′ is 7-dimensional, and coincides with V ∗ ( D A ). Note that E = T , and D = T , .Hence we have seen that the free and some of the nearly free curves give riseto smooth components of the analytic subspace V ead ( S , ..., S r ). Now we investigatewhen such components have the expected dimension, given by Theorem 3.2 (2). Theorem 3.8.
Let C be a reduced plane curves such that C ∈ V ead ( S , ..., S r ) . Thenthe analytic subspace V ead ( S , ..., S r ) is smooth at the point C of the expected dimen-sion d ( d + 3)2 − τ ( C ) if and only if mdr ( f ) ≥ d − .Proof. Using the first row in the commutative diagram (3.1), we see that there is anisomorphism H ( J f | C ( d )) = H ( T h C i ). Then, using Serre duality and the identityΩ(log C ) = Hom( T h C i , O P ) = T h C i ( d − h ( T h C i ) = h ( T h C i ( d −
6) = h ( E C ( d − ar ( f ) d − . (cid:3) Corollary 3.9.
Let C be a reduced plane curves such that C ∈ V ead ( S , ..., S r ) . Thenthe analytic subspace V ead ( S , ..., S r ) is smooth at the point C of the expected dimen-sion d ( d + 3)2 − τ ( C ) if the curve C satisfies one of the following conditions. (1) The degree d of C is at most 5. (2) The degree d of C is at most 6 and C is free and irreducible. (3) The curve C is a nodal curve. The third claim is known, see [29, 39], but our approach is perhaps new.
Proof.
The first claim follows from our assumption that we consider in this paper onlycurves with mdr ( f ) >
0. The second claim follows from the fact that an irreduciblefree curve C : f = 0 satisfy mdr ( f ) ≥
2, see [16, Theorem 2.8]. The third claimfollows from the fact that a nodal curve C : f = 0 satisfy mdr ( f ) ≥ d −
2, see [14,Theorem 4.1]. (cid:3)
Corollary 3.10.
Let C be a reduced plane curves such that the analytic subspace V ead ( S , ..., S r ) is smooth at the point C of the expected dimension d ( d + 3)2 − τ ( C ) . Then one has the following. (1)
If the curve C is free, then degree d of C is at most 7. (2) If the curve C is nearly free, then degree d of C is at most 8.Proof. It is enough to recall that for a free (resp. nearly free) curve C : f = 0 onehas mdr ( f ) < d/ mdr ( f ) ≤ d/ (cid:3) For a rational cuspidal plane curve C which satisfies Conjecture 2.7, we cantest whether C is ea -rigid by using Corollary 3.6, and test the fact that C is ea -unobstructed by using Theorem 3.8 and Corollary 3.10.The next example shows that for curves which are not (nearly) free, the differencebetween the orbit G · C and the scheme V ead ( S , ..., S r ) can be quite large. Example 3.11.
In this example we revisit some examples from [25, 31] and treatthem from our point of view.(1) The curve C : f = y ( x + 2 y + z )( x − y − z )( x − x z + y z + y z ) = 0is considered in [25, Example 5.3]. This curve has three triple points of type D and seven nodes A , hence τ ( C ) = 3 · V ea (3 D , A ) = V es (3 D , A ), since only simple singularities are involved. Greuel and Lossenshow that the scheme V ea (3 D , A ) is smooth at C of dimension 16. Thisresult follows also from Theorem 3.8, since one has d = 7 and mdr ( f ) = 5 ≥ d − C : f = x + z ( xz + y ) = 0 is considered in [25, Example5.6 (c)]. This curve is irreducible, has a unique singularity, which is of type A and was used by Luengo in [31] to construct the first singular variety EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 11 V ead ( S , ..., S r ). It is known that for this curve one has h (( J f | C )( d )) = 20and h (( J f | C )( d )) = 1, see [24, Examples 6.4 (6)]. We can recover thesedimensions as follows. For this polynomial f , one has, by direct computationusing SINGULAR, d = 9, n ( f ) = 12 and mdr ( f ) = 4. By Theorem 3.3 wehave h (( J f | C )( d )) = dim T C V ea ( A ) = 8 + 12 = 20 . From the proof of Theorem 3.8, we have that h (( J f | C )( d )) = ar ( f ) = 1 , since in a minimal set of generators for the graded S -module AR ( f ) there isonly one of degree 4 and three more generators, all having degree 8.(3) The curve C : f = x + x z + z ( xz + y ) = 0 is considered in [25, Example5.6 (c)]. This curve is irreducible, has a unique singularity, which is of type A and is a deformation of Luengo’s curve discussed above. Greuel andLossen show that the analytic subspace V ea ( A ) is smooth at C of dimension23. This result follows also from Theorem 3.8, since one has d = 9 and mdr ( f ) = 5 ≥ d − On the rational plane curves with at least 3 cusps
It is well known that a rational plane curve with at least 3 cusps has degree d ≥ Proposition 4.1.
There is only one rational plane curve with 3 cusps of degree d = 4 up to projective equivalence, namely the tricuspidal quartic C : [(2) , (2) , (2)] = [3 A ] , ( s t − s : s t : t − st ) . Up to projective equivalence, there are three rational cuspidal quintics having at least3 cusps. They have the following cuspidal configurations and parametrizations. (1) C : [(3) , (2 ) , (2)] = [ E , A , A ] , ( s t − s : s t : − st + t ) . (2) C ′ : [(2 ) , (2 ) , (2 )] = [3 A ] , ( s t − s : s t − s : − s − s t − st + t ) . (3) C ′′ : [(2 ) , (2) , (2) , (2)] = [ A , A ] , ( s t : s t − s : t + 2 s t ) . Using our algorithm to decide the freeness of a rational curve given by a parametriza-tion described in [3], we get the following.
Corollary 4.2.
The curve C is nearly free with exponents (2 , . The curves C , C ′ and C ′′ are free with exponents (2 , . Moreover, C , C and C ′′ have the followingequations C : f ( x, y, z ) = x y + y z + x z − xyz ( x + y + z ) = 0 , C : f = 9 xy − y − x y z + 48 xy z − y z + 16 x z = 0 , C ′′ : f = − x + 2 x y − x yz + y z − xy z + x z = 0 . The equation for C given above does not correspond to the given parametrization.The equation for C ′ obtained from the above parametrization is too complicated tolist.The rational cuspidal plane curves with 3 cusps of degree d ≥ F Z ( d, a ),is described in [22], see Theorem 3.5 and Proposition 3.9. Theorem 4.3.
Let C be a rational cuspidal curve of type ( d, d − with at least threecusps. Then C has exactly three cusps and there exists a unique pair of integers a, b , a ≥ b ≥ with a + b = d − such that the multiplicity sequences of the cusps are [( d − , (2 a ) , (2 b )] . Moreover, up to projective equivalence the equation of C = C d,a can be written in affine coordinates ( x, y ) as f ( x, y ) = x a +1 y b +1 − (( x − y ) d − − xyg ( x, y )) ( x − y ) d − , where d ≥ , g ( x, y ) = y d − h ( x/y ) and h ( t ) = X k =0 ,d − a k k ! ( t − k , with a = 1 , a = a − and a k = a ( a − · · · ( a − k + 1) for k > . Conjecture 4.4.
The curves C d,a ∈ F Z ( d, a ) are free divisors and mdr ( f ) = 2 forall possible values of the pair ( d, a ), when d ≥
5. In particular, τ ( C d,a ) = d − d + 7for any pair ( d, a ) with d ≥ d = 4 the curve C , is the quartic with 3 cusps A from Example 3.7 (3),and it is nearly free with exponents ( d , d ) = (2 , d, a ) with 5 ≤ d ≤ F Z ( k ), is described in [23], see Theorem 1.1. Theorem 4.5.
Let C be a rational cuspidal curve of type ( d, d − with at leastthree cusps and d ≥ . Then d = 2 k − where k ≥ , C has exactly three cuspswith multiplicity sequences [(2( k − , k − ) , (3 k − ) , (2)] . Moreover, up to projectiveequivalence, this curve C = C k is given by a parametrization ( s : t ) ( s k − t : s k − ( s − t ) (2 s + t ) : t ( s − t ) q k ( s, t )) , where q k ∈ C [ s, t ] is the homogeneous polynomial of degree k − defined below. The polynomial q is given in [23], see equation (4.1) below. The polynomials q = q k , for k >
4, are constructed as follows. We start with a polynomial f ∈ C [ x, y ],homogeneous of degree k −
5. We make the substitution x = t and y = t − t + 2,and we define a polynomial in t by the formula A ( t ) = f ( t , t − t + 2) + t k − g ( t ) , where g ( t ) is a polynomial in t of degree 2 k −
7. The polynomials f and g are uniquelydetermined by the following conditions.(1) The m -th derivative of A vanish at t = − ≤ m ≤ k − n -th derivative of A vanish at t = 1 for 0 ≤ n ≤ k − EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 13 (3) The polynomial in t defined by h ( t ) = A ( t )( t − t + 2) k − satisfies h (1) = − h ′ (1) = 3.Finally, we define q by asking that q (1 , t ) = t h ( t ) + 1( t − . Here are the first of these polynomials, q being given in [23] and q , q being com-puting using this approach.(4.1) q ( s, t ) = s + 2 st + 3 t . (4.2) q ( s, t ) = s + 2 s t + 3 s t + 3611 st + 2711 t . (4.3) q ( s, t ) = s + 2 s t + 3 s t + 612169 s t + 621169 s t + 468169 st + 243169 t . (4.4) q ( s, t ) = s +2 s t +3 s t + 37801009 s t + 41491009 s t + 39421009 s t + 31591009 s t + 19441009 st + 7291009 t . Conjecture 4.6.
The curves C k ∈ F Z ( k ) are free divisors and mdr ( f ) = k − k ≥
4. In particular, τ ( C d ) = 3( k − for any k ≥ ≤ k ≤ F E ( k ), is described in [21], see Theorem 1.1. For thedefinition of T V ( h D i ), see the final section. Theorem 4.7.
Let C be a rational cuspidal curve of type ( d, d − with at leastthree cusps, d ≥ and satisfying χ ( T V ( h D i ) ≤ . Then d = 3 k − where k ≥ , C has exactly three cusps with multiplicity sequences [(3( k − , k − ) , (4 k − , ) , (2)] and χ ( T V ( h D i ) = 0 . Moreover, up to projective equivalence, this curve C = C k isgiven by a parametrization ( s : t ) ( t k − ( t − s ) : s ˜ q k ( s, t ) : s t k − ) , where ˜ q k ∈ C [ s, t ] is the homogeneous polynomial of degree k − defined below. For the description of the polynomial ˜ q k we use the fomulas given in [2, MainTheorem (h)], following the suggestion kindly offered to us by Karol Palka. Definefirst a sequence of polynomials X k ∈ C [ u ] by the formulas X ( u ) = 2 u − u and X k +1 ( u ) = u ( X k ( u ) − X k (1)) u − . Note that deg X k = 3( k −
3) and X k is divisible by u . Then set Y k ( t ) = t k − X k ( 1 t ) , denote by ˜ Y k ( s, t ) the polynomial obtained by homogenization of the polynomial Y k ,and note that ˜ Y k ( s, t ) is divisible by s . Set ˜ q k = ˜ Y k ( s, t ) /s . For instance, we get inthis way the following parametrizations.(4.5) ˜ q ( s, t ) = − s + st + t . (4.6) ˜ q ( s, t ) = − s + s t + s t + st + t . (4.7) ˜ q ( s, t ) = − s − s t + s t + 2 s t + 3 s t + 3 s t + 3 st + 3 t . Conjecture 4.8.
The curves C k ∈ F E ( k ) are free divisors and mdr ( f ) = k − k ≥
5. In particular, τ ( C k ) = 7( k − − k + 3 for any k ≥ ≤ k ≤ Conjecture 4.9.
Any rational cuspidal plane curve C of degree d ≥ d , d ) determine the infiniteseries to which it belongs, namely(1) C ∈ F Z ( d, a ) if and only if d = 2, and in this case d = d − C ∈ F Z ( k ) if and only if d = ( d − / ≥
3, and in this case d = d .(3) C ∈ F E ( k ) if and only if d = ( d + 2) / ≥
4, and in this case d = (2 d − / d = mdr ( f ), so the above conjecture can be restated using mdr ( f ). Inview of formula (2.6), the above conjecture can be restated using the total Tjurinanumber τ ( C ) as well.4.10. On the rigidity conjecture.
We recall here the rigidity conjecture, following[22]. Let C ⊂ P be a reduced, irreducible curve and let V → P be the minimalembedded resolution of the singularities of C , such that the total transform D of C is a SNC-divisor in V . Let T V h D i be the logarithmic tangent bundle of the pair( V, D ). Then H ( T V h D i ) is the space of infinitesimal automorphisms of the pair( V, D ), H ( T V h D i ) is the space of the infinitesimal deformations of the pair ( V, D )and H ( T V h D i ) is the space of the obstructions for extending such infinitesimal de-formations. Moreover the deformations of the pair ( V, D ) corresponds to equisingulardeformations of C inside P . We say that C is FZ-projectively rigid if h ( T V h D i ) = 0and that is FZ-unobstructed if h ( T V h D i ) = 0. Flenner and Zaidenberg show in [22],see bottom of page 444, that in fact C is FZ-projectively rigid if and only if C is es -rigid as in our Definition 3.4. They continue by stating the following conjecture. EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 15
Conjecture 4.11 (Rigidity Conjecture) . A rational cuspidal curve C ⊂ P which hasat least 3 singularities is FZ-projectively rigid and FZ-unobstructed. In particular,such a curve is es -rigid.Flenner and Zaidenberg have checked in [22, 23] that this conjecture holds for theseries of cuspidal rational plane curves F Z ( d, a ) and F Z ( k ). One has also thefollowing conjecture, also due to Flenner and Zaidenberg. Conjecture 4.12 (Weak Rigidity Conjecture) . A rational cuspidal curve C ⊂ P which has at least 3 singularities satisfies χ ( T V h D i ) = 0.Fenske has shown in [21] that the curves in the series F E ( k ) satisfy the weakrigidity conjecture. A stronger conjecture, called the Negativity Conjecture, wasproposed by Palka [34, 35], and has extremely interesting consequences for the studyof rational cuspidal plane curves.On the other hand, our Conjecture 4.9 implies that any rational cuspidal planecurve C of degree d ≥ ea -rigid, by Corollary 3.6(1), but most of them are not ea -unobstructed by Corollary 3.10 (1).The results by Flenner, Zaidenberg and Fenske mentioned above imply that for acurve C in any of the families F Z ( d, a ), F Z ( k ) and F E ( k ), one has the equality G · C = V ead ( S , S , S ) = V esd ( S , S , S ) , because any equisingular deformation of C is in fact an equianalytic deformation.This equality holds also for all the curves in Proposition 4.1. Note that the equality V ead ( S , ..., S r ) = V esd ( S , ..., S r ) fails for rational cuspidal curves with one cusps, asExamples 5.2 and 5.4 below shows. In fact, the rational unicuspidal curves con-structed in Theorem 1.1 and Theorem 1.2 in [15] belong, for a fixed degree d , to aunique analytic subspace V esd ( S ), but to distinct analytic subspaces V ead ( S ), as theirTjurina numbers vary. Similar examples for curves with two cusps are also easy toproduce, for instance starting with the curve C : f = y + x z = 0. On the otherhand, the equality G · C = V ead ( S , ..., S r ) holds for rational cuspidal curves of degree d ≥ µ ( C ) = τ ( C ), see [10].5. Jacobian syzygies, Jacobian module and equisingular deformations
Recall the ideal sheaves I ( Z ∗ ( C )) introduced before Lemma 3.1. They enter intothe following exact sequence(5.1) 0 → I ( Z ea ( C ) | C ) → I ( Z es ( C ) | C ) → F C → , where the sheaf F is just the quotient o sheaf I ( Z es ( C ) | C ) / I ( Z ea ( C ) | C ). It followsthat the support of the sheaf F C is contained in the set of non-simple singularitiesof C , and at each such singularity p one has(5.2) dim F C,p = dim I es ( C, p ) /I ea ( C, p ) = m es ( C, p ) , as in (2.2). To get the value of this dimension in concrete cases, one may use thealgorithm described in [4] and implemented in SINGULAR [5], or Example 2.2 in very simple cases. As in Lemma 3.1, we have I ( Z ea ( C ) | C ) = J f | C , and hence weget the following long exact sequence of cohomology groups(5.3)0 → H ( J f | C ) → H ( I ( Z es ( C ) | C )) → H ( F C ) → H ( J f | C ) → H ( I ( Z es ( C ) | C )) → . This yields the following result.
Theorem 5.1.
Let C : f = 0 be a reduced plane curves such that C ∈ V ead ( S , ..., S r ) .Then the Zariski tangent space T C V esd ( S , ..., S r ) of the analytic subspace V esd ( S , ..., S r ) at the point C sits in the following exact sequence → T C V ead ( S , ..., S r ) → T C V esd ( S , ..., S r ) → H ( F C ) . Moreover, the morphism T C V esd ( S , ..., S r ) → H ( F C ) is surjective if mdr ( f ) ≥ d − .In particular, one has dim T C V ead ( S , ..., S r ) ≤ dim T C V esd ( S , ..., S r ) ≤ dim T C V ead ( S , ..., S r )+ X p m es ( C, p ) , and equality holds in the second inequality if mdr ( f ) ≥ d − . Moreover, the ana-lytic subspace V esd ( S , ..., S r ) is smooth of the expected dimension at the point C if mdr ( f ) ≥ d − . Here the sum P p m es ( C, p ) is over all the non-simple singularities of C . Thelast claim in this theorem follows from the formula (2.1), Theorem 2.5 (2) and theidentification H ( J f | C ) = AR ( f ) d − in the exact sequence (5.3), recall the proof ofTheorem 3.8. Example 5.2.
Consider the rational cuspidal curve C : f = y + x z = 0 . This curve has a unique singularity S at p = (0 : 0 : 1) with µ ( C, p ) = 12. TheJacobian ideal J f is generated by x , y , x z and the ideal I f is generated by x , y .It follows that h = x y ∈ I f , but h / ∈ J f , hence h yields a basis for N ( f ) . For t = 0, the curve C t : f t = f + th = y + x z + tx y = 0has again a singularity at p = (0 : 0 : 1), which is analytically isomorphic to thesingularity S . It follows that the deformation ( C t , p ) is equianalytic. If we applyTheorem 3.8, we see that the analytic subspace V ea ( S ) is smooth of the expecteddimension ed at C , where ed = 20 −
12 = 8. It is easy to see that all the curves C t for t = 0 are in the same G -orbit. Moreover dim G · C = 7, dim G · C = 8 and G · C is obviously contained in the closure of the orbit G · C . It follows that one has theequality V ea ( S ) = G · C ∪ G · C as germs at C . Note that C is a nearly free curve with mdr ( f ) = 1, while C : f = 0is a free curve with mdr ( f ) = 2.Now we turn our attention to the analytic subspace V es ( S ). Using Example 2.2,we see that the equisingular ideal I es of the singularity T , : u + v = 0 is spannedby u , v and u v , in particular m es ( C, p ) = 1. On the other hand, it is known that
EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 17 the tangent space T C ( G · C ) is given by the projectivization of J f, , and hence atransversal T to this tangent space is spanned by the following (cid:18) (cid:19) − dim J f, = 21 − x y , x y a z b for a, b ≥ a + b = 3, xy i z j for i ≥ j ≥ i + j = 4, y p z q , p ≥ q ≥ p + q = 5. Any deformation of C : f = 0 is given by a deformationin this transversal T , up-to the action of the group G . If we transform the abovemonomials in x, y, z into monomials in u, v by setting x = u , y = v and z = 1, andwe look for those monomials in u, v giving µ -constant deformations of the singularity T , , we get only the following two monomials x y and x y . The deformation in thedirection of the monomial x y was already discussed above. For t = 0, the curve C ′ t : f ′ t = y + x z + tx y = 0has again a singularity at p = (0 : 0 : 1), which is equisingular to the singularity S , since the deformation is µ -constant. On the other hand, this deformation is nolonger analytically trivial, since τ ( C ′ , p ) = 11 <
12 = τ ( C, p ) = µ ( C, p ) . Indeed, all the curves C ′ t for t = 0 are in the same G -orbit, and one can check that C ′ is nearly free with mdr ( f ) = 2.If we apply Theorem 5.1, we see that the analytic subspace V es ( S ) is smooth ofthe expected dimension ed at C , where ed = 20 −
12 + 1 = 9. It follows that we havethe following equality of smooth germs at C V es ( S ) = G · ( f + T ) , where T is the 2-dimensional vector subspace in the transversal T spanned by thetwo monomials x y and x y . In fact, note that the curve C α,β : f + αx y + βx y = 0is projectively equivalent to the curve C αt ,βt , for any t = 0. It follows that V es ( S ) = G · C ∪ [ α : β ] ∈ P (3 , G · C α,β , i.e. V es ( S ) is the union of the 7-dimensional orbit G · C and of a 1-parameter familyof 8-dimensional G -orbits, all of them containing G · C in their closure, parametrizedby the weighted projective line P (3 ,
2) = P . Remark 5.3.
Note that the deformation C ′ t above has essentially the same propertiesas the deformation C ′′ t : f ′′ t = y + x z + txy = 0in the direction of xy , a vector in the tangent space T C ( G · C ). Example 5.4.
Consider the rational cuspidal curve C : f = y + x z = 0 . This curve has a unique singularity S at p = (0 : 0 : 1) with µ ( C, p ) = 20. TheJacobian ideal J f is generated by x , y , x z and the ideal I f is generated by x , y . It follows that h = x y ∈ I f , but h / ∈ J f , hence h yields a basis for N ( f ) . For t = 0, the curve C t : f t = f + th = y + x z + tx y = 0has again a singularity at p = (0 : 0 : 1), which is equisingular to the singularity S ,but has τ ( C t , p ) = 19 <
20 = τ ( C, p ) = µ ( C, p ) . Hence a deformation given by a monomial in I f is not necessarily equianalytic as inExample 5.2 above. Note that all the curves C t for t = 0 are in the same G -orbit, C is a nearly free curve with mdr ( f ) = 1, while C : f = 0 is a free curve with mdr ( f ) = 2.If we apply Theorem 3.3, we see that dim T C V ea ( S ) = 8. Moreover, we know by[10, Theorem 1.1] that one has V ea ( S ) = G · C , and hencedim V ea ( S ) = dim G · C = 7 . Hence the analytic subspace V ea ( S ) is non-reduced at any point, even though itssupport V ea ( S ) is smooth. Note that in this example, unlike the classical exampleof B. Segre of curves with many simple cusps, discussed in [42] or in [37, Example4.7.10], the dimension of V ea ( S ) is the expected dimension d ( d + 3) / − τ ( C ) = 27 −
20 = 7 . Now we turn our attention to the analytic subspace V es ( S ). Using Example 2.2,we see that the equisingular ideal I es of the singularity T , : u + v = 0 is spannedby u , v , u v and u v , in particular m es ( C, p ) = 3. On the other hand, it is knownthat the tangent space T C ( G · C ) is given by the projectivization of J f, , and hencea transversal T to this tangent space is spanned by the following (cid:18) (cid:19) − dim J f, = 28 − x y , x y a z b for a, b ≥ a + b = 3, x y i z j for i, j ≥ i + j = 4, xy p z q , p ≥ q ≥ p + q = 5, and y m z n for m ≥ n ≥ m + n = 6. As above, anydeformation of C : f = 0 is given by a deformation in this transversal T , up-to theaction of the group G . If we transform the above monomials in x, y, z into monomialsin u, v by setting x = u , y = v and z = 1, and we look for those monomials in u, v giving µ -constant deformations of the singularity T , , we get only the following threemonomials x y , x y and x y .If we apply Theorem 5.1, we see that the analytic subspace V es ( S ) satisfies8 ≤ dim T C V es ( S ) < . Note that the curve C α,β,γ : f + αx y + βx y + γx y = 0is projectively equivalent to the curve C αt ,βt ,γt , for any t = 0. It follows that V es ( S ) = G · C ∪ [ α : β : γ ] ∈ P (4 , , G · C α,β,γ , EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 19 i.e. V es ( S ) is the union of the 7-dimensional orbit G · C and of a 2-parameter familyof 8-dimensional G -orbits, all of them containing G · C in their closure, parametrizedby the weighted projective plane P (4 , , V es ( S ) = 8 + 2 = 10and hence we get dim T C V es ( S ) ≤
10 = dim V es ( S ) . It follows that the analytic subspace V es ( S ) is smooth at C , and of the expecteddimension, which is ed = 27 −
20 + 3 = 10, although 1 = mdr ( f ) < d − P (4 , ,
2) = P (2 , , X = P (2 , ,
1) hastwo singular points, namely x = (1 : 0 : 0) and x = (0 : 1 : 0), see [11]. The curvecorresponding to x is the curve C introduced above. Using Theorem 5.1, it followsthat the analytic subspace V es ( S ) is smooth of the expected dimension ed at C ,where ed = 27 −
19 + 2 = 10, since m es ( C , p ) = 2. The curve corresponding to x is the curve C ′ : f ′ = f + x y = 0 , which is nearly free with mdr ( f ′ ) = 3 and τ ( C ′ , p ) = 18. Using Theorem 5.1, itfollows that the analytic subspace V es ( S ) is smooth of the expected dimension ed ′ at C ′ , where ed ′ = 27 −
18 + 1 = 10, since m es ( C ′ , p ) = 1. In this example theTjurina numbers τ ( C, p ) and τ es ( C, p ) and the Tjurina modalities m es ( C, p ) havebeen computed using the SINGULAR software, [5].
Remark 5.5.
Note that the deformation of the sextic C above given by C ′′ t : f ′′ t = y + x z + txy = 0in the direction of xy , a vector in the tangent space T C ( G · C ) produces nearly freecurves with mdr ( f ′′ t ) = 2 for t = 0. One can show by simple linear transformationsthat this deformation is G -equivalent to a equisingular deformation of C inside oneof the orbits G · C α,β,γ . References [1] E. Artal Bartolo, L. Gorrochategui, I. Luengo, A. Melle-Hern´andez, On some conjecturesabout free and nearly free divisors, in:
Singularities and Computer Algebra, Festschrift forGert-Martin Greuel on the Occasion of his 70th Birthday , pp. 1–19, Springer (2017) 2.6[2] M. Borodzik, H. ˙Zo l¸adek, Complex algebraic plane curves via Poincar-Hopf formula. II. Annuli.Israel J. Math. 175 (2010), 301–347. 4[3] L. Bus´e, A. Dimca, G. Sticlaru, Freeness and invariants of rational plane curves,arXiv:1804.06194. 4[4] A. Campillo, G.-M. Greuel, C. Lossen, Equisingular calculations for plane curve singularities,J. Symb. Comp. 42 (2007), 89–114. 5[5] W. Decker, G.-M. Greuel, G. Pfister and H. Sch¨onemann.
Singular [9] A. Dimca, Freeness versus maximal global Tjurina number for plane curves, Math. Proc.Cambridge Phil. Soc. 163 (2017), 161–172. 2.6, 2.6[10] A. Dimca, On rational cuspidal plane curves, and the local cohomology of Jacobian rings,arXiv:1707.05258. to appear in Commentarii Mathematici Helvetici. 2.6, 2.6, 4.10, 5.4[11] A. Dimca, S. Dimiev, On analytic coverings of weighted projective spaces, Bull. London Math.Soc. 17 (1985), 234–238. 5.4[12] A. Dimca, D. Popescu, Hilbert series and Lefschetz properties of dimension one almost completeintersections, Comm. Algebra 44 (2016), 4467–4482. 2.6[13] A. Dimca, E. Sernesi, Syzygies and logarithmic vector fields along plane curves, Journal del’´Ecole polytechnique-Math´ematiques 1(2014), 247-267. 2.6, 3[14] A. Dimca, G. Sticlaru, Koszul complexes and pole order filtrations, Proc. Edinburgh Math.Soc. 58 (2015), 333–354. 3[15] A. Dimca, G. Sticlaru, On the exponents of free and nearly free projective plane curves, Rev.Mat. Complut. 30(2017), 259–268. 4.10[16] A. Dimca, G. Sticlaru, Free divisors and rational cuspidal plane curves, Math. Res. Lett.24(2017), 1023–1042. 2.6, 3[17] A. Dimca, G. Sticlaru, Free and nearly free curves vs. rational cuspidal plane curves, Publ.RIMS Kyoto Univ. 54 (2018), 163–179. 2.6, 2.6, 3[18] A. Dimca, G. Sticlaru, On the freeness of rational cuspidal plane curves, arXiv:1802.06688, toappear in Moscow Math.J. 2.6[19] I. Dolgachev, Weighted projective varieties, in: Group Actions and Vector Fields, LNM 956(1982), pp. 34–71. 5.4[20] A.A. du Plessis and C.T.C. Wall, Curves in P ( C ) with 1-dimensional symmetry, Revista MatComplutense 12 (1999), 117–132. 2.6, 3[21] T. Fenske, Rational cuspidal curves of type ( d, d −
4) with χ (Θ V h D i ) ≤
0, Manuscripta Math.98 (1999), 511–527. 1, 4, 4.10[22] H. Flenner, M. Zaidenberg, On a class of rational plane curves, Manuscripta Math. 89 (1996),439–459. 1, 4, 4.10, 4.10[23] H. Flenner, M. Zaidenberg, Rational cuspidal plane curves of type ( d, d − EFORMATIONS OF PLANE CURVES AND JACOBIAN SYZYGIES 21 [34] K. Palka, Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture, J. ReineAngew. Math. (Crelle’s Journal), online 2016, 28 pp; arXiv:1405.5346 4.10[35] K. Palka, T. Pe lka, Classification of planar rational cuspidal curves. I. C**-fibrations, Proc.London Math. Soc. 115 (2017), 638–692. 4.10[36] J. Piontkowski, On the number of cusps of rational cuspidal plane curves, Experiment. Math.,16(2007), 251-255. 4[37] E. Sernesi, Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wis-senschaften 334, Springer-Verlag, Berlin, 2006. 1, 5.4[38] E. Sernesi, The local cohomology of the jacobian ring, Documenta Mathematica, 19 (2014),541-565. 1, 2.6, 3, 3, 3[39] F. Severi, Vorlesungen ¨uber Algebraische Geometrie, Anhang F. Leipzig: Teubner 1921. 1, 3[40] A. Simis, S.O. Toh˘aneanu, Homology of homogeneous divisors, Israel J. Math. 200 (2014),449-487. 2.6[41] D. van Straten, T. Warmt, Gorenstein duality for one-dimensional almost completeintersections–with an application to non-isolated real singularities, Math. Proc.Cambridge Phil.Soc.158(2015), 249–268. 2.6[42] A. Tannenbaum, On the classical characteristic linear series of plane curves with nodes andcuspidal points: two examples of Beniamino Segre, Compositio Mathematica, 51 (1984), 169–183. 1, 5.4[43] J. Wahl, Deformations of plane curves with nodes and cusps, Amer. J. Math. 96 (1974), 259–577. 1, 2.4[44] J. Wahl, Equisingular deformations of plane algebroid curves. Trans. Amer. Math. Soc.193(1974), 143–170. 1, 2.1
Universit´e Cˆote d’Azur, CNRS, LJAD, France
E-mail address : [email protected] Faculty of Mathematics and Informatics, Ovidius University Bd. Mamaia 124,900527 Constanta, Romania
E-mail address ::