Relations between tautological cycles on

We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Next we obtain a vanishing result for some of the generating classes pi; this gives an improvement of an earlier result of Herbaut. Finally we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk’s operator D lifts to the tautological subalgebra of CH(J). MSC Classification: 14C25, 14H40. Introduction 0.1. Let (J, θ) be the Jacobian of a curve C of genus g > 2, and let j: C →֒ J be the embedding obtained by choosing a point x0 ∈ C. The Chow ring CH(J) (with Q-coefficients) comes equipped with a number of structures: in addition to the usual intersection product we have the Pontryagin product ∗, the Fourier transform F , and the action of n and n∗ for all n ∈ Z. Using the latter we obtain Beauville’s decomposition CH(J) = ⊕CH(j)(J). These structures are inherited by the quotient A(J) := CH(J)/ ∼alg. In [2], Beauville studied the tautological subring T (C) ⊂ A(J), that can be defined as the smallest subalgebra that is stable under all operations just mentioned, and that contains the image of A(C) under j∗. He proved that T (C) is generated by certain classes pm that are the components of F [C] in the Beauville decomposition. This leads to the question what is the ideal of relations between these classes. The work of Polishchuk [12] provides us with a powerful method to produce relations. He considers the map φ: R → T (C) from the polynomial ring R := Q[x1, x2, . ] given by xi 7→ pi. The elements of R that are of degree > g (with xi of degree i) with zero coefficient in front of x g 1 lie in the kernel of φ for obvious reasons. But now the point is that Polishchuk is able to give an explicit differential operator D on R that preserves the kernel of φ, and that the ideal of “trivial” relations is far from stable under D . Polishchuk studies the smallest ideal I ⊂ R that is D-stable and that contains all trivial relations. Conjecturally, for general curves C (over a base field of sufficiently high transcendence degree) the quotient ring R := R/I maps isomorphically to the tautological ring T (C). The present paper has three main goals: (1) First we want to give some new results on the structure of the ring R. Though in principle it is not difficult to calculate, for a given g, the dimensions of all summands R (j) on a computer, it is not so easy to obtain general conclusions about the structure of R from Polishchuk’s methods. (In some sense the main obstacle is that the variable x1 plays an exceptional role in the calculations.) 1 We prove a theorem, Thm. 2.6, that gives more insight in the structure of R. Each R(j) := ⊕i R i (j) is a finite-dimensional sl2-module. We give a simple recipe for the multiplicity of a given irreducible sl2module in each R(j). As corollaries we get some general results about the relations in R(j), especially for small and large (relative to g) values of j. Further we present a conjecture of van der Geer and Kouvidakis that gives a simple recipe for the dimensions of the spaces R (j). (2) The second goal of the paper is to investigate cycle relations for curves with a given linear system. If the curve has a g d, Herbaut [8] proved that we get new relations between the classes pm. This result was reproved and simplified by van der Geer and Kouvidakis in [7]. However, the ideal generated by I together with these new relations is not stable under the operator D . Ideally we should like to get a full description of the quotient of R by the ideal generated by Herbaut’s relations together with all their images under powers of D . This turns out to be not so easy. However, we do obtain a vanishing result, Thm. 3.7 for classes pk with k > d + 1 − 2r. It turns out that the result we obtain is a slight improvement of an earlier result of Herbaut in [8]. (3) The third goal of the paper is to lift some results obtained thus far to the Chow ring of J . For general curves Polishchuk already obtained a number of results about this in [13]. He considered a subalgebra taut(C) ⊂ CH(J) that we call the small tautological ring, and he proved that is generated by classes pm (lifting those considered above) and qm (essentially the Beauville components of F (j∗K), with K the canonical class on C). In itself, it is not very hard to lift the result of Herbaut and van der Geer-Kouvidakis to the Chow level. We do this in Section 3, following the method of [7] which is based on a Grothendieck-RiemannRoch calculation. See Thm. 5.6 for the result. However, if we also want to lift the operator D to the Chow level, we find that the “small” tautological ring needs to be enlarged, as the calculations involve natural classes that are in general not contained in it. So we define the big tautological ring Taut(C) ⊂ CH(J) as the smallest Qsubalgebra that is stable under all operations ·, ∗, F , n and n∗ and that contains the image of j∗: CH(C) → CH(J). Again there is an operator D , but as Taut(C) is not finitely generated it is now less easy to describe this operator explicitely. However, the calculations we should like to do involve only finitely many classes of the form F (j∗Di) with Di divisors on C. We show that the subalgebra of Taut(C) generated by these classes together with the pm and qm is stable under D and that D again acts on it as a differential operator that we can give explicitely. Closely related results have been obtained independently by Alexander Polishchuk; see Remark 0.3 below. The situation we end up with is that we have lifted Herbaut’s relations to the Chow level, and that we also have an explicit operator D . Combined this gives an abundance of relations that can easily be calculated on a computer. We give some examples of the relations thus obtained in the final section. What seems much harder is to get some control over the total ideal of relations thus obtained. In particular, at this moment we have no analogue of Thm. 2.6 for curves with a given linear system. 0.2. Acknowledgement. I thank Tom Koornwinder for his help getting started with Mathematica. I thank Gerard van der Geer and Alexis Kouvidakis for allowing me to present their conjecture on the dimensions of the spaces R (j), see 2.13, and for explaining to me some details pertaining to their conjecture. 0.3. Remark. Very recently, some related results have appeared. In particular we should like to draw the reader’s attention to Polishchuk’s preprint [14], in which some ideas appear that are closely 2 related to our Section 4, and to the preprint [6] of Fu and Herbaut, that contains some results related to the material in Section 5. 0.4. Notation and conventions. We work over a fixed algebraically closed field k. Throughout, if X is a non-singular complete variety then by CH(X) we denote the Chow ring of X tensored with Q. We write sl2 = Q · f + Q · h+ Q · e with [e, f ] = h, [e, h] = −2e, and [f, h] = 2f . §1. The Chow ring of a Jacobian 1.1. Let C be a non-singular complete curve of genus g > 2. Let J := PicC/k be its Jacobian, θ: J ∼ −→ J t the canonical polarization. We also write θ ∈ CH(J) for the class of a symmetric theta-divisor. Choose a base point x0 ∈ C, and let j: C →֒ J be the embedding given on points by x 7→ [