Points in projective spaces and applications

Abstract

We prove the factoriality of a nodal hypersurface in P 4 of degree d that has at most 2(d − 1) 2 /3 singular points, and factoriality of a double cover of P 3 branched over a nodal surface of degree 2r having less than (2r − 1)r singular points.We assume that all varieties are projective, normal, and defined over C.

On a conjecture of Ciliberto

“…Let C be a curve in P 4 of degree λ. Then|Supp(C) ∩ Sing(X)| λ(d − 1), and if |Supp(C) ∩ Sing(X)| = λ(d − 1), then Sing(C) ∩ Sing(X) = ∅.See the proof in[8], Lemma 29.It follows from[13] that the following conditions are equivalent:-the hypersurface X is factorial; -the points of the set Sing(X) impose independent linear conditions on hypersurfaces in P 4 of degree 2d − 5. Suppose that |Sing(X)| (d − 1) 2 and the hypersurface X contains no planes.…”

“…The aim of this paper is to give an independent geometric proof of Theorem 1.6, which is based on the results obtained in [8] and [9]. Our paper has the following structure: in § 2 we consider some auxiliary results; in § 3 we prove Theorem 3.1, which is used in the proof of Theorem 1.6; in § 4 we prove Theorem 1.6 omitting the proof of Lemma 4.10; in § 5 we prove Lemma 4.10.…”

Maximal families of nodal varieties with defect

“…Consider now hypersurfaces containing a fixed plane P. They form a family of codimension 1 2 (d + 1)(d + 2). Since the Grassmannian of planes in P 4 has codimension 6 it follows that the total family has codimension 1 2 (d 2 + 3d − 10). A general element of this family is of the form 1 f 1 + 2 f 2 with deg( i ) = 1 and deg( f i ) = d − 1.…”

Which quartic double solids are rational?

“…This is implied by the following result. Theorem 1.8 ([15], [5,Theorem 6]). A nodal quartic double solid with at most seven nodes is Q-factorial unless it has exactly six nodes and is described by Example 1.4.…”