Poincar\'e bundle for the fixed determinant moduli space on a nodal curve

TL;DR

This paper proves the stability of the Poincaré bundle on a nodal curve's fixed determinant moduli space, advancing understanding of vector bundles in singular algebraic geometry.

Contribution

It establishes the stability of the Poincaré bundle on the fixed determinant moduli space over a nodal curve, including restrictions to points and the entire space.

Findings

The restriction of the Poincaré bundle to a point is stable.

The Poincaré bundle on the entire moduli space is stable with respect to certain polarizations.

Stability holds for both the fixed determinant moduli space and the space of vector bundles with a fixed determinant.

Abstract

Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction $\mathcal{U}_{L,x}$ of the Poincare bundle to $x \times U_Y(n,d,L)$ is stable with respect to the polarisation $\theta_L$ and its restriction to $x \times U'_Y(n,d,L)$, where $U'_Y(n,d,L)$ is the moduli space of vector bundles of rank $n$ and determinant $L$, is stable with respect to any polarisation. We show that the Poincar\'e bundle $\mathcal{U}_{L}$ on $Y \times U_Y(n,d,L)$ is stable with respect to the polarisation $a \alpha + b \theta_L$ where $\alpha$ is a fixed ample Cartier divisor on $Y$ and $a, b$ are positive integers.