On the lifting problem in ${\mathbb{P}}^4$ in characteristic $p$

Given ${\mathbb{P}}_k^4$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset {\mathbb{P}}_k^4$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0{\mathcal{I}}_Y(s)\mathrm{\ne }0$ and $h^0{\mathcal{I}}_X(s)=0$ for some $s>0$. This determines a nonzero element $\alpha \in H^1{\mathcal{I}}_X(s)$ such that $\alpha \cdot H=0$ in $H^1{\mathcal{I}}_X(s)$. We find different upper bounds of $d$ in terms of $s$, $p$ and the order of $\alpha$ and we show that these bounds are sharp. In particular, we see that $d\le s^2$ for $p<s$ and $d\le s^2-s+2$for$p\ge s$.