Asian Journal of Mathematics

Determinant Line Bundles on Moduli Spaces of Pure Sheaves on Rational Surfaces and Strange Duality

Yao Yuan

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Abstract

Let $M^H_X (u)$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u = (0, L, \xi(u) = 0)$, i.e. sheaves in $u$ are of dimension 1. There is a natural morphism $\pi$ from the moduli space $M^H_X (u)$ to the linear system $|L|$. We study a series of determinant line bundles $\lambda_{c_n^r}$ on $M^H_X (u)$ via $\pi$. Denote $g_L$ the arithmetic genus of curves in $|L|$. For any $X$ and $g_L \le 0$, we compute the generating function $Z^r(t) = \sum_n h^0(M^H_X (u), \lambda_{c_n^r})t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (−e))$ with $e = 0, 1$, we compute $Z^1(t)$ for $g_L \gt 0$ and $Z^r(t)$ for all $r$ and $g_L = 1, 2$. Our results provide a numerical check to Strange Duality in these specified situations, together with Göttsche’s computation. And in addition, we get an interesting corollary (Corollary 4.2.13) in the theory of compactified Jacobian of integral curves.

Article information

Source
Asian J. Math., Volume 16, Number 3 (2012), 451-478.

Dates
First available in Project Euclid: 23 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1353696017

Mathematical Reviews number (MathSciNet)
MR2989230

Zentralblatt MATH identifier
1262.14013

Subjects
Primary: 14D22: Fine and coarse moduli spaces 14J26: Rational and ruled surfaces

Keywords
line bundle strange duality conjecture moduli spaces of semistable sheaves

Citation

Yuan, Yao. Determinant Line Bundles on Moduli Spaces of Pure Sheaves on Rational Surfaces and Strange Duality. Asian J. Math. 16 (2012), no. 3, 451--478. https://projecteuclid.org/euclid.ajm/1353696017


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