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

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.