Congruences of lines in P5\mathbb{P}^5, quadratic normality, and completely exceptional Monge-Amp\`ere equations

The existence is proved of two new families of locally Cohen-Macaulay sextic threefolds in $\mathbb{P}^5$, which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Amp\`ere equations. One of these families comes from a smooth congruence of multidegree $(1,3,3)$ which is a smooth Fano fourfold of index two and genus 9.Comment: 16 page

Vector spaces of skew-symmetric matrices of constant rank

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2r and type (N+1) 7(N+1) under the natural action of SL(N+1), over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1-generic matrices of linear forms. We also show that, for each rank two vector bundle on P^2 defining a triple Veronese embedding of P^2 in G(1,7), there exists a vector space of 8 78 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle

Low degree 3-folds in \bold P\sp 6

In this paper we construct new smooth varieties of dimension 3 in P^6 with degree 12 =< d =< 15, and we also give different constructions for some known varieties. Moreover we determine the adjunction theoretic structure of all the varieties that we deal with

Erratum to "On the Hilbert scheme of Palatini threefolds"

We prove the following theorem, which corrects a previous result of the authors (M. L. Fania, E. Mezzetti, On the Hilbert scheme of Palatini threefolds. Adv. Geom. 2 (2002), 371\u2013389): Let D be a web of linear complexes in P^5 such that D is contained in the dual of G (1, 5). Then the natural rational map from G(3,P^14) to the irreducible component of the Hilbert scheme of threefolds in P^5, containing the Palatini scrolls, is not regular at the point corresponding to D, unless D is contained in the tangent space to G(3, 5) at a point

Semipolarized nonruled surfaces with sectional genus two

Complex projective nonruled surfaces S endowed with a numerically effective line bundle L of arithmetic genus g(S,L)=2 are investigated. In view of existing results on elliptic surfaces we focus on surfaces of Kodaira dimension k(S)=0 and 2. Structure results are provided in both cases according to the values of L^2. When S is not minimal we describe explicitly the structure of any birational morphism from S to its minimal model S', reducing the study of (S,L) to that of (S',L'), where L' is a numerically effective line bundle with g(S',L')=2 or 3. Our description of (S,L) when S is minimal, as well as that of the pair (S',L') when g(S',L')=3 relies on on several results concerning linear systems, mainly on surfaces of Kodaira dimension zero. Moreover, several examples are provided, especially to enlighten the case in which S is a minimal surface of general type, (S,L) having Iitaka dimension 1

Mukai varieties as hyperplane sections

Let (\sM,\sL) be a smooth $(n+1)$-dimensional variety polarized by an ample and spanned line bundle \sL. Let $A$ be a smooth member of |\sL|. Assume that $n\geq 4$ and that $(A,H_A)$ is a Mukai variety, i.e., $-K_A\approx (n-2)H_A$ for some ample line bundle $H_A$ on $A$. Let $H$ be the line bundle on \sM which extends $H_A$. We show that \sM is a Fano variety and either $H$ is ample, in which case the cones of effective $1$-cycles $NE(A)$ and NE(\sM) on $A$ and \sM coincide, or \grk(H)=n, $H$ is semiample and (\sM,\sL) has a structure of a conic fibration. Then most of the paper is devoted to classify the pair (\sM,\sL) in the case when \sL is very ample