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

TL;DR

This paper investigates the lifting problem for integral surfaces in projective 4-space over fields of positive characteristic, establishing sharp bounds on the degree based on cohomological conditions and characteristic parameters.

Contribution

It provides new upper bounds on the degree of surfaces in characteristic p, depending on cohomological data and the order of a specific cohomology class, with proofs of sharpness.

Findings

For p<s, degree d ≤ s^2.

For p ≥ s, degree d ≤ s^2 - s + 2.

Bounds are proven to be sharp.

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 $p\ge s$.