Remarks on the defining equations of smooth threefolds in \({\mathbb P}^5\)
The authors are interested in the vanishing of \(H^1(\mathbb P^n, {\mathcal I}_X (t))\), for \(t\) small, when \(X\) is a smooth threefold in \(\mathbb P^5\) over the complex numbers. (When such vanishing occurs, \(X\) is said to be \(t\)-normal.) In particular, the authors are motivated by the following conjecture of Peskine and Zak: Let \(X\) be a nondegenerate, not necessarily smooth, projective variety of codimension \(e\) in \(\mathbb P^n\). Then (1) \(H^i(\mathbb P^n, {\mathcal I}_X(j)) = 0\) for \(i \geq 1, j \geq 0, i+j < \dim(X)/(e-1)\); (2) For \(i \geq 1\), \(j \geq 0\), \(i+j = \dim(X)/(e-1)\), it is possible to describe all varieties for which \(H^i(\mathbb P^n, {\mathcal I}_X(j)) \neq 0\). It has also been conjectured by Peskine and Van de Ven that all smooth threefolds in \(\mathbb P^5\) are 2-normal except the Palatini scroll of degree 7. On the other hand, \textit{N. Yu. Netsvetaev} [in: Topology and geometry, Rohlin Semin. 1984-1986, Lect. Notes Math. 1346, 433--453 (1988; Zbl 0664.14028)] and \textit{A. Arsie} [\texttt{http://arxiv.org/abs/math.AG/0005142}] have results of the type that if \(X\) has low codimension and can be defined by a small number of equations then in fact it is a complete intersection (and so in particular those cohomology modules vanish). In this paper, the authors first give some examples that show the sharpness of Arsie's result. Their main result, however, is that if \(X\) is a smooth threefold in \(\mathbb P^5\) cut out scheme theoretically by at most four equations then \(X\) is \(t\)-normal for \(t \leq 3\). Furthermore, if a technical (but natural) surjectivity condition holds then they also conclude that \(X\) is 4-normal. They do this by studying the cohomology of a certain rank three vector bundle on \(\mathbb P^5\) associated to \(X\).
scientific article; zbMATH DE number 1901491