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Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces

Purnaprajna Bangere, Jayan Mukherjee, Debaditya Raychaudhury

TL;DR

The paper studies Ulrich bundles on smooth projective varieties through the Koszul property, the semistability of iterated syzygy bundles, and the rationality of moduli spaces on Del Pezzo surfaces. It connects the deformation theory of Ulrich bundles to their syzygies, establishes stability criteria via curve-reduction techniques, and proves that on anticanonically embedded Del Pezzo surfaces with $d\ge4$ every Ulrich bundle satisfies the Koszul property and is slope-semistable; it further shows that iterated syzygies give birational correspondsences between moduli spaces and yield infinitely many rational moduli spaces for higher rank. The results provide new evidence for Costa–Miró-Roig’s rationality conjecture in this setting and demonstrate density of iterated syzygy bundles inside the corresponding moduli. Overall, the work advances understanding of the interplay between Ulrichness, Koszulity, syzygies, and moduli on Del Pezzo surfaces, with concrete descriptions in low-degree cases including cubic surfaces.

Abstract

Let $\mathcal{E}$ be a vector bundle on a smooth projective variety $X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section $H$. In this article, we study the Koszul property of $\mathcal{E}$, the slope-semistability of the $k$-th iterated syzygy bundle $\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.

Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces

TL;DR

The paper studies Ulrich bundles on smooth projective varieties through the Koszul property, the semistability of iterated syzygy bundles, and the rationality of moduli spaces on Del Pezzo surfaces. It connects the deformation theory of Ulrich bundles to their syzygies, establishes stability criteria via curve-reduction techniques, and proves that on anticanonically embedded Del Pezzo surfaces with every Ulrich bundle satisfies the Koszul property and is slope-semistable; it further shows that iterated syzygies give birational correspondsences between moduli spaces and yield infinitely many rational moduli spaces for higher rank. The results provide new evidence for Costa–Miró-Roig’s rationality conjecture in this setting and demonstrate density of iterated syzygy bundles inside the corresponding moduli. Overall, the work advances understanding of the interplay between Ulrichness, Koszulity, syzygies, and moduli on Del Pezzo surfaces, with concrete descriptions in low-degree cases including cubic surfaces.

Abstract

Let be a vector bundle on a smooth projective variety that is Ulrich with respect to the hyperplane section . In this article, we study the Koszul property of , the slope-semistability of the -th iterated syzygy bundle for all and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if is a Del Pezzo surface of degree , then any Ulrich bundle satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters , the corresponding moduli spaces of slope-stable bundles when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.
Paper Structure (12 sections, 33 theorems, 43 equations, 2 tables)

This paper contains 12 sections, 33 theorems, 43 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The relations between TEXT and TEXT
  3. Semistability for (iterated--)syzygy bundles of Ulrich bundles
  4. (Semi--)Stability of vector bundles
  5. (Semi--)Stability for syzygy bundles of Ulrich bundles
  6. Koszul property and iterated syzygy bundles
  7. Consequences on the moduli of vector bundles on Del Pezzo surfaces
  8. Chern classes of vector bundles
  9. Deformations and moduli of vector bundles
  10. Moduli of vector bundles on Del Pezzo surfaces
  11. Moduli of stable bundles on cubic surfaces
  12. Moduli of stable bundles on Del Pezzo surfaces of degree $d\geq 4$

Key Result

Theorem 1.1

Let $X_d\subseteq\mathbb{P}^d$ be the anti--canonically embedded Del Pezzo surface of degree $3\leq d\leq 8$. Let $\mathcal{E}$ be a stable Ulrich vector bundle with respect to $H$ of rank $r$ where $H:=\mathcal{O}_{\mathbb{P}^d}(1)|_{X_d}=-K_{X_d}$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 34 more