arXiv subjects

Ben Moonen

Publications and source records attributed to Ben Moonen.

23 records · Page 2Linked to original sources

arXiv2009

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t] , the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivations with regard to t and u act. Our results give rise to a new grading on the Chow ring of the Jacobian. After tensoring with Q the associated descending filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's. Finally we give a version of our main result for tautological classes, and we show how our methods give a very simple and geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis.

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arXiv2009

Divided powers in Chow rings and integral Fourier transforms

We prove that for any monoid scheme M over a field with proper multiplication maps from M x M to M, we have a natural PD-structure on the ideal CH_{>0}(M) of CH(M) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to 2^N-torsion, where N is the integral part of 1 + log_2(3g). As a consequence we obtain, over an algebraically closed field, a PD-structure (for the intersection product) on 2^N A, where A is the augmentation ideal of CH(J). We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.

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arXiv2007

Relations between tautological cycles on Jacobians

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 p_i; 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 Chow(J).

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arXiv2002

Serre-Tate Theory for Moduli Spaces of PEL Type

The main goal of this paper is to generalize Serre-Tate theory of "ordinary" local moduli to Shimura varieties of PEL type. To this end we develop a generalized notion of ordinariness, we prove a number of basic results about this, and we study the formal deformations of ordinary objects. In general, the formal deformation spaces get a new "group-like" structure that we call a "cascade". Further the paper contains some results on the Ekedahl-Oort stratification of PEL moduli spaces, and an application of our results to the study of congruence relations.

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arXiv2002

A dimension formula for Ekedahl-Oort strata

We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by cosets in a Weyl group. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.

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