{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:4LLHYAKZ4LMWN7LT2HOTEXQWDD","short_pith_number":"pith:4LLHYAKZ","schema_version":"1.0","canonical_sha256":"e2d67c0159e2d966fd73d1dd325e1618f9ad4e9f98638a66ebc5e2aea02cf113","source":{"kind":"arxiv","id":"1109.1738","version":1},"attestation_state":"computed","paper":{"title":"On the lifting problem in $\\mathbb P^4$ in characteristic $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Paola Bonacini","submitted_at":"2011-09-08T15:06:26Z","abstract_excerpt":"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 $p0$, 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