On the Moduli Space of Coherent Systems of Type $(2, c_1, c_2, 2)$ on Projective Plane

TL;DR

This paper investigates the moduli space of certain coherent systems on the projective plane, providing conditions for their existence, nonemptiness, and analyzing their topological properties.

Contribution

It introduces new numerical criteria for the existence and nonemptiness of moduli spaces of coherent systems of a specific type on P^2, and studies their topological flips.

Findings

Necessary conditions for $ ext{α}$-semistability are established.

Numerical conditions for the nonemptiness of the moduli space are derived.

Topological properties of flips are analyzed.

Abstract

We study the moduli space of coherent systems in $P^2$ using the Segre invariant. We obtain necessary conditions for the existence of $\alpha$-semistable coherent systems $(E,V)$ of type $(2, c_1, c_2, k)$, with $k \geq 2$. Afterwards, we give numerical conditions to the nonemptiness of the moduli space and compute the critical values depending of the Chern classes. Finally, we give some topological properties of the flips.