Abstract
For abelian surfaces of Picard rank 1, we perform explicit computations of the cohomological rank functions of the ideal sheaf of one point, and in particular of the basepoint-freeness threshold. Our main tool is the relation between cohomological rank functions and Bridgeland stability. In virtue of recent results of Caucci and Ito, these computations provide new information on the syzygies of polarized abelian surfaces.
Highlights
Throughout this note we work over an algebraically closed field K.Motivated by the continuous rank functions of Barja, Pardini and Stoppino [1], in their paper [10] Jiang and Pareschi introduced the cohomological rank functions hiF,l associated to a coherent sheafF on a polarized abelian variety (A, l)
[10] Jiang and Pareschi introduced the cohomological rank functions hiF,l associated to a coherent sheaf on a polarized abelian variety (A, l)
Throughout this section, (S, l) will be a polarized abelian surface satisfying the hypothesis of Theorem A, namely: l is of type (1, d), and for every divisor class D we have l2|D · l
Summary
Throughout this note we work over an algebraically closed field K. Motivated by the continuous rank functions of Barja, Pardini and Stoppino [1], in their paper [10] Jiang and Pareschi introduced the cohomological rank functions hiF,l associated to a coherent sheaf (or more generally, a bounded complex of coherent sheaves). F on a polarized abelian variety (A, l). Makes sense of the i-th (hyper)cohomological rank of F twisted with a (general) representative of the fractional polarization xl.
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