Alberto Alzati

Special subhomaloidal systems of quadrics and varieties with one apparent double point

We classify smooth n-dimensional varieties Xn ⊂ P2n+1 with one apparent double point and of degree d 6 2n + 4, showing that these are only the smooth irreducible divisors of type (2, 1), (0, 2) and (1, 2) on the Segre manifold P1 × P ⊂ P2n+1, a 3-fold of degree 8 and two Mukai manifolds, the first one of dimension 4 and degree 12, the second one of dimension 6 and degree 16. We also prove that a linearly normal variety Xn ⊂ P2n+1 of degree d 6 2n + 1 and with Sec(Xn) = P2n+1 is regular and simply connected, that it has one apparent double point and hence it is a divisor of type (2, 1), (0, 2) or (1, 2) on the Segre manifold P1 × P ⊂ P2n+1. To this aim we study linear systems of quadrics on projective space whose base locus is a smooth irreducible variety and we look for conditions assuring that they are (completely) subhomaloidal; we also show some new properties of varieties Xn ⊂ P2n+1 defined by quadratic equations and we study projections of such varieties from (subspaces of ) the tangent space.

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 THEOREM OF MATHER ON GENERIC PROJECTIONS IN THE SETTING OF ALGEBRAIC GEOMETRY

We present the proof of the theorem of Mather on generic projections, stated in the setting of algebraic geometry. The main tools used are the Thom-Boardman singularities in the jet space. This theorem has been applied in the study of codimension two submanifold ofPn and it seems that it could have further applications.

A Geometric Approach to the Trifocal Tensor

In geometric computer vision the trifocal tensors are 3×3×3 tensors T by whose means three different camera views of the same scene are related to each other. In this paper we find two different sets of constraints, in the entries of T, that must be satisfied by trifocal tensors. The first set gives exactly the (closure of the) trifocal locus, i.e. all trifocal tensors, but it is very big. The second set, although not complete and still very big, has the property that it is possible to extract from it a set of only eight equations that are generically complete, i.e. for a generic choice of T, they suffice to decide whether T is indeed trifocal. Note that 8 is the codimension of the trifocal locus in its ambient space.

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.

Linear pose estimate from corresponding conics

We propose here a new method to recover the orientation and position of a plane by matching at least three projections of a conic lying on the plane itself. The procedure is based on rearranging the conic projection equations such that the non linear terms are eliminated. It works with any kind of conic and does not require that the shape of the conic is known a-priori. The method was extensively tested using ellipses, but it can also be used for hyperbolas and parabolas. It was further applied to pairs of lines, which can be viewed as a degenerate case of hyperbola, without requiring the correspondence problem to be solved first. Critical configurations and numerical stability have been analyzed through simulations. The accuracy of the proposed algorithm was compared to that of traditional algorithms and of a trinocular vision system using a set of landmarks.

Computing camera focal length by zooming a single point

In this paper we present a novel simple procedure to compute the focal length of a camera. The method is based on zooming in and out only a single point. The same approach allows computing the principal point when only two points are available on a pair of images surveyed with a different focal length. Experimental results show that the method is as accurate as classical full calibration methods. Moreover, its application to augmented reality produces more accurate results than those obtained when the simple pin-hole model is considered.

Some extremal contractions between smooth varieties arising from projective geometry

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective

Quadro-Quadric Special Birational Transformations of Projective Spaces

n

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

-dimensional varieties, with