On codimension-two subcanonical varieties inside $\mathbb{P}^n$

TL;DR

This paper investigates conditions under which codimension-two subcanonical varieties in projective space are complete intersections, establishing criteria involving deficiency modules and demonstrating that certain Buchsbaum varieties are complete intersections.

Contribution

It provides new sufficient conditions based on deficiency modules for codimension-two subcanonical varieties to be complete intersections and proves that specific Buchsbaum varieties are also complete intersections.

Findings

Sufficient conditions on deficiency modules imply a variety is a complete intersection.

Smooth 3-Buchsbaum varieties in high-dimensional projective space are complete intersections.

Criteria involving cohomology and ideal sheaf sections are established.

Abstract

Let $X \subseteq \mathbb{P}^n, n \geq 4$ be a codimension-two subcanonical local complete intersection variety with ideal sheaf $\mathcal{I}_X$. Let $a_X \in \mathbb{Z}$ be such that $\omega_X = \mathscr{O}_X(a_X)$. Assume that there exists $\displaystyle j \leq \frac{a_X+n+2}{2}$ such that $\Gamma(\mathcal{I}_X(j)) \neq 0$. We prove some sufficient conditions on the first deficiency module $\mathrm{H}^1_*(\mathcal{I}_X)$ that ensures that $X$ is a complete intersection. We also show that smooth codimension-two $3$-Buchsbaum varieties inside $\mathbb{P}^n, n \geq 6$ are complete intersections.