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A lower bound on the Ulrich complexity of hypersurfaces

Angelo Felice Lopez, Debaditya Raychaudhury

TL;DR

The paper establishes sharp lower bounds for the Ulrich complexity of smooth hypersurfaces of degree $d\ge3$ in projective space. It develops a degeneracy-locus construction for a globally generated Ulrich bundle $\mathcal{E}$ of rank $r$, and analyzes the associated locus $Z=D_1(\varphi)$ to derive stringent numerical constraints via Riemann–Roch in low dimensions. By exhaustively treating ranks $r\in\{4,5,6,7\}$ and relevant dimensions (notably $n=6$ and $n=8$ in the very general case), the authors obtain contradictions that rule out the existence of small-rank Ulrich bundles, thereby proving new lower bounds: $\operatorname{Uc}(X)\ge6$ when $n=7$ or $n=6$ very general, and $\operatorname{Uc}(X)\ge8$ when $n\ge9$ or $n=8$ very general. The work combines detailed Chern-class calculations for Ulrich bundles with precise invariants of degeneracy loci, leveraging vanishing results and Riemann–Roch to connect geometry with Ulrich complexity.

Abstract

We give a lower bound on the Ulrich complexity of hypersurfaces of dimension $n \ge 6$.

A lower bound on the Ulrich complexity of hypersurfaces

TL;DR

The paper establishes sharp lower bounds for the Ulrich complexity of smooth hypersurfaces of degree in projective space. It develops a degeneracy-locus construction for a globally generated Ulrich bundle of rank , and analyzes the associated locus to derive stringent numerical constraints via Riemann–Roch in low dimensions. By exhaustively treating ranks and relevant dimensions (notably and in the very general case), the authors obtain contradictions that rule out the existence of small-rank Ulrich bundles, thereby proving new lower bounds: when or very general, and when or very general. The work combines detailed Chern-class calculations for Ulrich bundles with precise invariants of degeneracy loci, leveraging vanishing results and Riemann–Roch to connect geometry with Ulrich complexity.

Abstract

We give a lower bound on the Ulrich complexity of hypersurfaces of dimension .

Paper Structure

This paper contains 9 sections, 27 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. notation and conventions
  3. A useful degeneracy locus
  4. Generalities on Ulrich vector bundles
  5. Invariants of hypersurfaces and their Ulrich bundles
  6. Proof of Theorem \ref{['main']}
  7. Chern classes of exterior powers
  8. General Riemann-Roch calculations
  9. Riemann-Roch calculations on hypersurfaces

Key Result

Theorem 1

Let $X \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d \ge 3$. Then the following lower bounds hold:

Theorems & Definitions (57)

  • Theorem 1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 47 more