A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

Abstract

AbstractWe study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$({\mathbb T},\ell )$\end{document}. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb T}$\end{document} and show that no smooth genus 5 curve on such a surface can contain a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$g^1_3$\end{document}. We also describe explicitly the singular divisors in the linear system |2ℓ|.