Existence of Ulrich Bundle on general Surfaces

TL;DR

This paper investigates the existence and stability of Ulrich bundles on general smooth projective surfaces with specific geometric properties, expanding understanding of their algebraic and geometric structures.

Contribution

It establishes conditions for the existence and stability of Ulrich bundles on surfaces with Picard rank one and ample canonical bundle, a novel contribution in algebraic geometry.

Findings

Ulrich bundles exist under certain geometric conditions.

Stability properties of these bundles are characterized.

Provides new insights into the structure of algebraic surfaces.

Abstract

Let $X$ be a smooth projective algebraic surface of Picard rank one with very ample canonical bundle $K_X$. We further assume that $q -1 \le \chi(\mathcal{O}_X$. In this article, we will study the existence of the Ulrich bundle and its stability property of it with respect to $K_X$.