Marino Palleschi

Discriminant loci of varieties with smooth normalization

Let X be a smooth complex projective n-fold endowed with an ample and spanned line bundle (L). Under the assumption that Γ(L) defines a generically one-to-one map we describe the singular set of the general element in the main component of the discriminant locus of |L|. This description is used to show that (X:,L) is covered by linear Pk’s, where k + 1 stands for the codimension of the main component. We also give some applications relating k to the spectral value of (X, L) and discuss some examples.

On threefolds with low sectional genus

The general problem of rebuilding the threefolds X endowed with a given ample divisor H , possibly non-effective, is closely related to the study of the complete linear system | K X + H | adjoint to H . Many powerful results are known about | K X + H |, for instance when the linear system | H | contains a smooth surface or, more particularly, when H is very ample (e.g. see Sommese [S1] and [S2]). From this point of view we study some properties of | K X + H |, which turn out to be very useful in the description of the threefolds X polarized by an ample divisor H whose arithmetic virtual genus g ( H ) is sufficiently low.

Double covers of

The classical subject of surfaces containing a hyperelliptic curve (here a double cover of P 1 ) among their hyperplane sections was settled some years ago by the third author and Van de Ven [SV] (see also [Se], [Io]). This paper is devoted to answering the following general question arising very naturally from that problem.

{\bf P}^n

Let X be a complex projective manifold and A⊂X a non-singular hypersurface which is an ample divisor having characteristic cycles Ai non-singular in every dimension i⩾0. The pairs (X,A) such that g(A1)=h1,0 (X) are characterized.

as very ample divisors

Let E be a very ample vector bundle of rank n - 1 over a smooth complex projective variety X of dimension n > 3. The structure of (X,e) being known when κ(K X + det e) < 0, we investigate the structure of the adjunction mapping when 0 < κ(K X + det e) < n.

Characterizing Projective Bundles by Means of Ample Divisors.

The statement of Lemma (0.6) has to be corrected as follows, according to what is really proven. Sing(A) is pure k-dimensional and possesses a possibly non reduced Cohen-Macaulay structure. There is a gap in Lemma (2.2). The assertion that PðKÞXnU ! D0 is dominant is unproven. As a consequence, the proof of Theorem (2.3) shows the weaker statement that a general element A2D0 has a k-dimensional singular set Sing(A) of which one component is a linear P whose general point is a non-degenerate quadratic singularity. Using this weaker statement of Theorem (2.3), Lemma (2.4) and Corollary (2.5) remain true. Unfortunately, the proofs of the results in Sec. 3 no longer work, and in fact the example given below shows that the results are not true as stated.