Sheaf theoretic compactifications of the space of rational quartic plane curves

TL;DR

This paper constructs a sheaf theoretic compactification of the space of rational quartic plane curves using $ ext{alpha}$-semistable pairs, analyzing its birational relations and computing its Poincaré polynomial.

Contribution

It introduces a novel sheaf theoretic compactification of $R_4$ via $ ext{alpha}$-semistable pairs and explores its birational transformations through wall-crossings.

Findings

Derived the Poincaré polynomial of the compactified space

Established birational relations via wall-crossings

Provided a sheaf theoretic framework for rational quartic curves

Abstract

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincar\'e polynomial of the compactified space.