On quasi-complete intersections of codimension $2$

In this paper, we prove that if $X\subset\mathbb{P}^n$, $n\ge 4$, is a locally complete intersection of pure codimension $2$ and defined scheme-theoretically by three hypersurfaces of degrees $d_1\ge d_2\ge d_3$, then $H^1(\mathbb{P}^n,\mathcal{I}_X(j))=0$ for $j<d_3$ using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold $X\subset\mathbb{P}^5$ is projectively normal if $X$ is defined by three quintic hypersurfaces.