Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves

Abstract

Abstract. In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hernández Ruipérez and Tejero Prieto.

Higher rank Brill—Noether theory and coherent systems open questions

“…for all integers a. In particular, the birational type of G(α; n, d, k) depends only on n mod d. The following question is raised in [58].…”

“…The main question remaining is the analogue of Question 6.3. For some computations in this direction, covering also the fixed determinant case, see [58]. In particular, the Hodge polynomial is computed for all G i (2 + ad, d, 1).…”

Higher rank Brill-Noether theory and coherent systems: Open questions

“…There is a very interesting use of Fourier-Mukai transforms in [HT08], which shows that G 0 This is proved to be true when gcd(n, d) = 1 and in some other cases in [LN09].…”

“…For some computations in this direction, covering also the fixed determinant case, see [LN09]. In particular, the Hodge polynomial is computed for all G i (2 + ad, d, 1).…”

Coherent systems on a nodal curve of genus one

“…The moduli spaces of coherent systems on smooth curves of small genera have been studied by Lange and Newstead in a series of papers. The case g = 0 (i.e., curve \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$X= {\mathbb P}^1)$\end{document} was studied in 17–19, while 20, 21 deal with g = 1. In this paper we consider coherent systems on an irreducible nodal curve of arithmetic genus 1 i.e., on the Weierstrass nodal curve.…”