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April, 2016 Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$
Jinwon CHOI, Kiryong CHUNG
J. Math. Soc. Japan 68(2): 685-709 (April, 2016). DOI: 10.2969/jmsj/06820685


We study the wall-crossing of the moduli spaces $\boldsymbol{M}^\alpha (d,1)$ of $\alpha$-stable pairs with linear Hilbert polynomial $dm+1$ on the projective plane $\mathbb{P}^2$ as we alter the parameter $\alpha$. When $d$ is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces $\boldsymbol{M}(d,1)$ of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan–Hölder filtrations is three.


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Jinwon CHOI. Kiryong CHUNG. "Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$." J. Math. Soc. Japan 68 (2) 685 - 709, April, 2016.


Published: April, 2016
First available in Project Euclid: 15 April 2016

zbMATH: 1342.14021
MathSciNet: MR3488141
Digital Object Identifier: 10.2969/jmsj/06820685

Primary: 14D20

Keywords: Betti numbers , blow-up/down , semistable pairs , wall-crossing formulae

Rights: Copyright © 2016 Mathematical Society of Japan


Vol.68 • No. 2 • April, 2016
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