## Journal of Differential Geometry

### Strange duality on rational surfaces

Yao Yuan

#### Abstract

We study Le Potier’s strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank $2$ sheaves with trivial first Chern class and second Chern class $2$, and the moduli space of $1$-dimensional sheaves with determinant $L$ and Euler characteristic $0$. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $\Sigma_e := \mathbb{P} (\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (e))$ except for $e = 1$. When $e = 1$, our result applies to $L = aG + bF$ with $b \geq a + [a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2 = -1$ and $[a/2]$ is the integral part of $a/2$.

#### Article information

Source
J. Differential Geom., Volume 114, Number 2 (2020), 305-336.

Dates
First available in Project Euclid: 1 February 2020

https://projecteuclid.org/euclid.jdg/1580526017

Digital Object Identifier
doi:10.4310/jdg/1580526017

Mathematical Reviews number (MathSciNet)
MR4058964

Zentralblatt MATH identifier
07163293

Subjects
Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)

#### Citation

Yuan, Yao. Strange duality on rational surfaces. J. Differential Geom. 114 (2020), no. 2, 305--336. doi:10.4310/jdg/1580526017. https://projecteuclid.org/euclid.jdg/1580526017