Ulrich ranks of Veronese varieties and equivariant instantons

TL;DR

This paper resolves the Ulrich-bundle rank classification for Veronese threefolds ${X_d}$, providing a complete description of ${\mathrm{Ur}}(X_d)$ depending on $d\bmod 6$. The authors construct Ulrich bundles as deformations of symmetric squares $S^2{\mathcal E}$ of $\mathrm{SL}_2$-equivariant instanton bundles ${\mathcal E}$ on ${\mathbb P}^3$, and relate these objects to orthogonal instantons via an étale map between moduli spaces. A key technical advance is proving $H^*(S^2{\mathcal E}_m(d-2))=0$ for the equivariant rank-2 instantons ${\mathcal E}_m$, enabling the formation of Ulrich bundles ${S^2{\mathcal E}}(2(d-1))$ on ${X_d}$. The results confirm Costa and Miró-Roig's conjecture in this setting and provide the first complete Ulrich-rank classification for a dimension greater than two Veronese variety, using a blend of representation theory, cohomology vanishing, and moduli-theoretic deformation theory.

Abstract

We construct Ulrich bundles on Veronese threefolds of arbitrary degree as generic deformations of symmetric squares of equivariant instanton bundles on the projective space, thus classifying the rank of Ulrich bundles on such varieties and proving a conjecture of Costa and Mir{ó}-Roig.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Remark 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['main-2']}