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Generic flatness of the cohomology of thickenings

Yairon Cid-Ruiz, Anurag K. Singh

TL;DR

The paper studies the generic flatness of cohomology for thickenings X_t of a closed subscheme X ⊂ P^n_A over a Noetherian base A. It develops relative Ampleness theory for vector bundles (f-ample bundles), establishes a uniform vanishing for Sym^t(N_X) that extends classical results, and uses Ext/local cohomology techniques to propagate flatness across all thickenings in a single dense open subset of Spec A when X → Spec A is smooth. The authors prove the conjecture on generic freeness in the setting of thickenings, and obtain a special case for ideals of m points with m ≤ n+2, which implies constant behavior of the least degrees α_t for generic point configurations in projective space. Together, these results provide a robust framework connecting deformation-theoretic phenomena, local cohomology, and the geometry of thickenings, with concrete implications for questions about symbolic powers and point configurations.

Abstract

Given a set of $m$ distinct points in projective space over a field, and $t$ a positive integer, a classical question asks for the least degree of a hypersurface that passes through each point with multiplicity at least $t$. Related to this, it remains unresolved whether there exists a dense open set of $m$-tuples of points for which this least degree is constant for each $t\ge 1$. We formulate a conjecture regarding the generic freeness of certain local cohomology modules that would answer this in the affirmative; we then prove this conjecture for up to $n+2$ points in projective $n$-space over a field. In a related direction, we prove a generic flatness result for the cohomology of thickenings of a closed subscheme of projective space over an integral domain.

Generic flatness of the cohomology of thickenings

TL;DR

The paper studies the generic flatness of cohomology for thickenings X_t of a closed subscheme X ⊂ P^n_A over a Noetherian base A. It develops relative Ampleness theory for vector bundles (f-ample bundles), establishes a uniform vanishing for Sym^t(N_X) that extends classical results, and uses Ext/local cohomology techniques to propagate flatness across all thickenings in a single dense open subset of Spec A when X → Spec A is smooth. The authors prove the conjecture on generic freeness in the setting of thickenings, and obtain a special case for ideals of m points with m ≤ n+2, which implies constant behavior of the least degrees α_t for generic point configurations in projective space. Together, these results provide a robust framework connecting deformation-theoretic phenomena, local cohomology, and the geometry of thickenings, with concrete implications for questions about symbolic powers and point configurations.

Abstract

Given a set of distinct points in projective space over a field, and a positive integer, a classical question asks for the least degree of a hypersurface that passes through each point with multiplicity at least . Related to this, it remains unresolved whether there exists a dense open set of -tuples of points for which this least degree is constant for each . We formulate a conjecture regarding the generic freeness of certain local cohomology modules that would answer this in the affirmative; we then prove this conjecture for up to points in projective -space over a field. In a related direction, we prove a generic flatness result for the cohomology of thickenings of a closed subscheme of projective space over an integral domain.
Paper Structure (4 sections, 10 theorems, 69 equations)

This paper contains 4 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Ample vector bundles relative to a base scheme
  3. Generic flatness in the smooth setting
  4. Ideals of points

Key Result

Theorem A

Let $A$ be a Noetherian domain containing a field of characteristic zero. Let $X\subset\mathbb{P}^n_A$ be a closed subscheme that is smooth over $\operatorname{Spec} A$, and $X_t$ its $t$-th thickening, as defined above. Fix an integer $j$, and set $\mathcal{F}\colonequals\mathcal{O}_{\mathbb{P}^n_A is a flat $A_a$-module for each integer $i\ge 0$, for all thickenings $t\ge 1$.

Theorems & Definitions (22)

  • Theorem A
  • Conjecture 1
  • Theorem B
  • Definition 1
  • Theorem 2.1
  • proof
  • Lemma 1
  • proof
  • Theorem 2.2
  • proof
  • ...and 12 more