On existence of Ulrich sheaf

TL;DR

Let $X$ be a smooth projective variety carrying an Ulrich bundle with respect to $O_X(1)$. The paper constructs an Ulrich sheaf on the $2$-nd symmetric power $Sym^{2}X$ (and extends to higher symmetric powers) by passing through the finite quotient map $\pi:X\times X\to Sym^{2}X$ and a descended ample line bundle $N$ with $\pi^{*}N \cong O_X(1)\boxtimes O_X(1)$. This yields Ulrich bundles on the nested Hilbert schemes $Z_n(C)$ and on Hillb^{n}C for curves, and proves the existence of Ulrich bundles on the blow-up $Bl_{A\times\{0\}}(A\times A)$ when $A$ carries an Ulrich bundle. The approach combines stability under products, descent under finite quotients, and cohomological vanishing criteria for Ulrichness under blow-ups to extend Ulrich existence to singular and birationally modified varieties. These results broaden the class of spaces known to admit Ulrich objects and illustrate effective use of pushforward, restriction, and descent techniques in Ulrich theory.

Abstract

Let X be a smooth projective variety carrying an Ulrich bundle. In the first part of this note, we construct an Ulrich sheaf on n-th symmetric power of X, which is a singular variety when $DimX >1$. As a consequence, we get the existence of an Ulrich bundle on Hillb^{n}C, where C is a smooth projective curve. Let A be an abelian variety which carries an Ulrich bundle. In the second part of this note, we show the existence of Ulrich bundle on the blow up of $A \times A$ along $A \times \{0\}$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5