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

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