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

Abstract

Given $\mathbb P^4_k$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset \mathbb P^4_k$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0\mathscr I_Y(s)\ne 0$ and $h^0\mathscr I_X(s)=0$ for some $s>0$. This determines a nonzero element $\alpha\in H^1\mathscr I_X(s)$ such that $\alpha\cdot H=0$ in $H^1\mathscr I_X(s)$. 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 $p\ge s$.