Brodmann, M. Cohomology of surfaces \(X\subseteq\mathbb{P}^r\) with degree \(\leq 2r-2\). (English) Zbl 0951.14011 Van Oystaeyen, Freddy (ed.), Commutative algebra and algebraic geometry. Proceedings of the Ferrara meeting in honor of Mario Fiorentini on the occasion of his retirement, Ferrara, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 206, 15-33 (1999). From the paper: Let \(K\) be an algebraically closed field and let \(X\subseteq\mathbb{P}^r_K\) be a non-degenerate projective variety of dimension \(>1\). It is known that, at least if \(X\) is of dimension \(\geq 6\), it may occur that \(h^1 (X,{\mathcal O}_X(-1))\neq 0\), even if \(X\) is smooth. So, the vanishing theorems of Kodaira and Mumford (which both hold if \(\text{char} (K)=0)\) may be hurt in a bad way if \(K\) is of positive characteristic. On the other hand it is still open, whether \(h^1(X,{\mathcal O}_X (n))=0\) for all \(n<0\) if \(X\) is a normal surface. In the present paper we show that the latter question has a positive answer if the degree of \(X\) is not too large, e.g. if \(d:=\deg (X)\leq 2r-2\). Assume that \(X\) has at most finitely many non-normal points and that \(\deg (X)\leq 2r-2\). We show that \(h^1(X,{\mathcal O}_X(n))\) takes the same value \(e^1(X)\) for all \(n<0\), that \(h^1(X, {\mathcal O}_X)\geq h^1(X, {\mathcal O}_X(1))\) and that \(h^1(X, {\mathcal O}_X(n+1)) \leq\max\{0,h^1 (X,{\mathcal O}_X(n))-1\}\) for all \(n>0\). We distinguish three classes of surfaces \(X\) which may occur, one of them being the well studied class of “quasi-\(K3\)-surfaces”.For the entire collection see [Zbl 0913.00044]. Cited in 1 Document MSC: 14F25 Classical real and complex (co)homology in algebraic geometry 14J25 Special surfaces 14N05 Projective techniques in algebraic geometry 14F17 Vanishing theorems in algebraic geometry Keywords:quasi-\(K3\)-surfaces; vanishing theorems; positive characteristic × Cite Format Result Cite Review PDF