On the Ulrichness of twisted syzygies and dual syzygies bundles

TL;DR

The paper classifies when twisted dual syzygy bundles $M^{\vee}_{\mathcal{L},V}\otimes \mathcal{L}^{k+1}$ (and related $M_{\mathcal{L},V}\otimes \mathcal{L}^{k-1}$) are Ulrich with respect to powers of a very ample line bundle $\mathcal{L}^a$ on a projective variety $X$. Using Euler sequences, identifications with ambient tangent bundles, and standard Ulrich criteria, it derives precise, dimension-dependent results: on curves the only possibilities occur for $X=\mathbb{P}^1$ with $L=\mathcal{O}(2)$ or $\mathcal{O}(1)$ (giving $a=k+2$ or $a=k+3$ respectively); on surfaces the unique case is $X=\mathbb{P}^2$ with $L=\mathcal{O}(2)$ yielding $T\mathbb{P}^2$ as Ulrich; for higher dimensions, the twisted dual syzygy bundles are not Ulrich. The paper also studies Ulrichness with respect to arbitrary polarizations $H$, obtaining partial results and concrete counterexamples via dual Ulrichs and Reider-type arguments. These findings deepen the link between syzygy geometry and Ulrich theory, providing a clear classification in low dimensions and showing obstructions in higher dimensions.

Abstract

Given a projective variety $X$ and a very ample line bundle $\mathcal{L}$ on $X$, we classify for which $X$ and $\mathcal{L}$ the twisted syzygies and twisted dual syzygies bundles are Ulrich with respect to the polarizations $\mathcal{L}^a$. We obtain some partial results when considering an arbitrary polarization $H$.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Proposition 3.4