Gian Mario Besana

Cell decomposition of almost smooth real algebraic surfaces

Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.

Projective normality of varieties of small degree

The projective normality of linearly normal smooth complex varieties of degree d ≤ 8 is investigated. The complete list of non projectively normal such manifolds is given; all of them are shown to be not 2-normal.

THE DIMENSION OF THE HILBERT SCHEME OF SPECIAL THREEFOLDS

The Hilbert scheme of 3-folds in ℙ n , n ≥ 6 , that are scrolls over ℙ 2 or over a smooth quadric surface Q ⊂ ℙ 3 or that are quadric or cubic fibrations over ℙ 1 is studied. All known such threefolds of degree 7 ≤ d ≤ 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.

Criteria for very ampleness of rank two vector bundles over ruled surfaces

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree. Dipartimento di Matematica, Univ. di Milano, 20133 Milano, Italy e-mail: alberto.alzati@unimi.it College of Computing and Digital Media, De Paul University, Chicago, IL, 60604, U.S.A e-mail: gbesana@depaul.edu Received by the editors May 21, 2008; revised February 6, 2009. Published electronically August 18, 2010. This work is within the framework of the national research project “Geomety of Algebraic Varieties” Cofin 2006 of MIUR. AMS subject classification: 14E05, 14J30.

Numerical criteria for very ampleness of divisors on projective bundles over an elliptic curve

In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a sufficient condition for a line bundle associated with a divisor D to be normally generated on

On the Dimension of the Adjoint Linear System for Quadric Fibrations

X=P(E)

Two dimensional scrolls contained in quadric cones in ℙ5

where E is a vector bundle over a smooth curve C. A line bundle which is ample and normally generated is automatically very ample. Therefore the condition found in Butlers work, together with Miyaokas well known ampleness criterion, give a sufficient condition for the very ampleness of D on X. This work is devoted to the study of numerical criteria for very ampleness of divisors D which do not satisfy the above criterion, in the case of C elliptic. Numerical conditions for the very ampleness of D are proved,improving existing results. In some cases a complete numerical characterization is found.

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

Let L^ be a very ample line bundle on a smooth, n-dimensional, projective manifold X^ , i.e. assume that \({L^ \wedge } \approx {i^*}{O_{pn}}\) (1) for some embedding \(i:{X^ \wedge } \to {\mathbb{P}^N}\). In [S1] it is shown that for such pairs, (X^ , L^), the Kodaira dimension of \({K_{{X^ \wedge }}} \otimes {L^{ \wedge n - 2}}\;is \geqslant 0\), i.e. there exists some positive integer, t, such that \({h^0}\left( {{{\left( {{K_{{X^ \wedge }}}{L^{ \wedge n - 2}}} \right)}^t}} \right) \geqslant 1\), except for a short list of degenerate examples. It is moreover shown that except for this short list there is a morphism \(r:{X^ \wedge } \to X\) expressing X^ as the blow-up of a projective manifold X at a finite set B, and such that: 1. \({K_{{X^ \wedge }}} \otimes {L^{ \wedge n - 1}} \approx {r^*}\left( {{K_X} \otimes {L^{n - 1}}} \right)\) where \(L: = {\left( {{r_*}{L^ \wedge }} \right)^{**}}\) is an ample line bundle and \({K_X} \otimes {L^{n - 1}}\) 2. \({K_X} \otimes {L^{n - 2}}\) is nef, i.e. \(\left( {{K_X} \otimes {L^{n - 2}}} \right) \cdot \geqslant 0\) for every effective curve C ⊂ X.

Peculiar loci of ample and spanned line bundles

Smooth, complex, ruled surfaces embedded in ℙ5 as linearly normal scrolls, such that they are contained in a quadric cone, are considered. Rational scrolls and some elliptic scrolls are shown to be the only ones contained in cones of rank 5. Results on scrolls contained in cones of lower ranks are also obtained.

On the k-regularity of some projective manifolds

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e = 0; 1 the scrolls ll up the entire component of the Hilbert scheme, while for e = 2 the scrolls exhaust a subvariety of codimension 1: