Abstracts

Marco Andreatta, University of Trento, Italy PDF

Title: Effective Adjunction Theory

Abstract: Let X be a projective variety and H a Cartier divisor on X. The effectivity, or non effectivity, of some adjoint divisors aK_X + bH, for suitable a,b, determines the geometry of X.I will first give a proof of the following version of the Termination of Adjunction: X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have H^0(X,m_0K_X +H) = 0 for some m_0 = m_0(H) > 0. Then I will discuss the following Conjecture: Assume that X has terminal singularities, H is nef and big and s > 0. H^0(X,K_X +tH) = 0 for every integer t with 1≤t≤s if and only if K_X +sH is not pseudoeffective; this is true if and only if the pair (X,L) is birational to a precise list of (uniruled) models. The Conjecture is true for s ≥ (dimX − 1); this can be proved via the Theory of the Reductions, started by Fujita and Sommese, which nowadays can be interpreted as a Minimal Model Program with Scaling.

Lucian Badescu University of Genova, Italy PDF

Title: Infinitesimal extensions of rank two vector bundles of submanifolds of small codimension

Abstract: Let X be a submanifold of dimension n of the complex projective space P^N (n < N), and let E be a vector bundle of rank two on X. If n >= (N+3)/2 >= 4 we prove a geometric criterion for the existence of an extension of E to a vector bundle on the first order infinitesimal neighbourhood of X in P^N in terms of the splitting of the normal bundle sequence of Y in X in P^N, where Y is the zero locus of a general section of a high twist of E. In the last section we show that the universal quotient vector bundle on the Grassmann variety G(k,m) of k-dimensional linear subspaces of P^m, with m >=3 and 1<= k <= m-2 (i.e. with G(k,m) not a projective space), embedded in any projective space P^N does not extend to the first infinitesimal neighbourhood of G(k,m) in P^N as a vector bundle.

Cinzia Casagrande, University of Torino, Italy

Title: Fano 4-folds with rational fibrations

Abstract: Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. Here we focus on Fano 4-folds with large second Betti number b_2, studied via birational geometry and the detailed study of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction. The main result that we want to present is the following: let X be a Fano 4-fold having a rational contraction X --> Y of fiber type (with dim Y > 0). Then either X is a product of surfaces, or b_2(X) is at most 12, or Y is P^1 or P^2.

Fabrizio Catanese, University of Bayreuth, Germany

Title: The double point formula with isolated singularities, and canonical embeddings

Abstract: Severi established in 1902 the double point formula for the number of improper double points of a variety X of dimension equal to the codimension. This was done in terms of projective invariants, called ceti. The modern version is given (in Hartshorne’s book without reference to Severi) in terms of the Chern classes of the tangent bundle of the normalization of X and of the hyperplane divisor. The main motivation for looking for generalizations is the following theorem, obtained with Keiji Oguiso (the result when X is smooth was obtained in my AMS 1995 Santa Cruz article): „ If a surface has geometric genus 5 and its canonical map is an embedding of the canonical model X, then X is a complete intersection of type (2,4) or (3,3)“ After explaining the classical versions of the double point formulae, and their relation, I shall present several generalizations, some over the complex numbers, some char free. I shall moreover briefly overview recent results concerning canonical surfaces in projective spaces of dimension 4 and 5. I shall then illustrate more general embedding obstructions derived from the double point formula, and shall deal with the problem of canonical embeddings of ample hypersurfaces in Abelian varieties.

Ciro Ciliberto, Univerisity of Roma Tor Vergata, Italy

Title: Irreducibility and unirationality of moduli spaces of polarized Enriques surfaces

Abstract: Abstract: Moduli spaces of polarized Enriques surfaces have several components, even if one fixes the degree of the polarization. In this talk I will discuss results concerning detecting irreducibile components and proving their unirationality. This is work in progress in collaboration with Th. Dedieu, C. Galati and A. Knutsen.

Alessio Corti, Imperial College London, UK

Title: Volume-preserving birational maps of 3-fold Mori fibred Calabi-Yau pairs

Abstract: I will discuss work in progress with Araujo and Massarenti concerning volume-preserving birational maps out of pairs (PP^3, B) where B is a nodal quartic.

Tommaso de Fernex, University of Utah, USA

Title: A simplicity criterion for normal isolated singularities

Abstract: The link of an isolated singular point of a complex variety is an analytic invariant of the singularity. It is natural to ask how much information the link carries about the singularity; for instance, the link of a smooth point is a sphere, and one can ask whether the converse is true. Work of Mumford and Brieskorn has shown that this is the case for normal surface singularities but not in higher dimensions. Recently, McLean asked whether more structure on the link may provide a way to characterize smooth points. In this talk, I will discuss how CR geometry can be used to define a link-theoretic invariant of singularities that distinguishes smooth points. The proof relies on a partial solution to the complex Plateau problem.

Sandra Di Rocco, KTH, Sweden PDF

Title: Polar Geometry, generalizations and applications

Abstract: this talk will be an attempt at conveying the importance of projective geometry in more applied settings. Polar classes are very classical objects in projective geometry. They have recently been rediscovered in connection with interesting applications of algebraic methods in imaging and variety sampling. I will try to explain these connections and give some new generalizations. The talk is partly based on joint work with David Eklund, Kathlen Kohn and Madeleine Weinstein.

Paltin Ionescu, University of Ferrara, Italy

Title: A remark on boundedness of manifolds embedded with small codimension

Abstract: For embedded projective manifolds of small codimension we give an explicit bound for their degree, depending on the Castelnuovo--Mumford regularity of their structure sheaf. As an application, we obtain bounds for the degree of such manifolds whose structure sheaf is arithmetically Cohen--Macaulay (in a weak sense) and whose canonical map is not birational.

Hoering Andreas Université de Nice Sophia Antipolis, France

Title: Positivity of cotangent sheaves for projective varieties with trivial canonical class

Abstract: Let X be a complex projective manifold. If X is not covered by rational curves, a theorem of Miyaoka tells us that the restriction of the cotangent sheaf to a sufficiently positive general complete intersection curve is nef. In this talk I will explain why this theorem can not be improved for projective manifolds with trivial canonical class. This "negative result“ has an interesting consequence : a Beauville-Bogomolov decomposition for varieties with kit singularities. This is based on joint work with Thomas Peternell.

Andreas Leopold Knutsen, University of Bergen, Norway

Title: Moduli of curves on Enriques surfaces

Abstract: Curves on K3 surfaces have been studied a lot and play a relevant role in the study of the moduli space of curves. By contrast, not so much is known about curves on Enriques surfaces, except that the rich geometry of the surface (in particular the existence of many elliptic pencils) make curves on Enriques surfaces quite special from the point of view of Brill-Noether theory. Moreover, the existence of a non-trivial 2-torsion bundle on the surface (the canonical bundle) endows every curve on an Enriques surface with a natural structure of a Prym curve. I will report on joint work in collaboration with C. Ciliberto, Th. Dedieu and C. Galati, where we in particular prove that, with a few exceptions, a general Prym curve lying on an Enriques surface lies on a unique such surface. The exceptions seem to be related to the existence of Enriques-Fano threefolds and to existence of Prym curves with nodal Prym-canonical model.

Antonio Lanteri, University of Milano, Italy PDF

Title: Characterizing some polarized manifolds via Hilbert curves

Abstract: Given a polarized manifold (X,L), consider the polynomial p(x,y):= \chi (xK_X+yL).The Hilbert curve of (X,L) is the complex affine algebraic plane curve \Gamma: = \Gamma_{(X,L)} defined by p(x,y)=0, when we look at x and y as complex variables. A natural expectation is that properties of (X,L) are encoded by its Hilbert curve. In fact, \Gamma is sensitive with respect to fibrations of X induced by a suitable adjoint linear system to the polarizing line bundle L. In particular, if X is a projective bundle over a smooth curve it turns out that \Gamma has the shape of a comb, and, conjecturally, this special shape characterizes the structure of (X,L). I will give a sketch of progress on this conjecture (done with Andrea L. Tironi) and discuss further properties of \Gamma for some special varieties arising in adjunction theory.

Carla Novelli, University of Padova, Italy

Title: Fano varieties with small non-klt locus

Abstract: Let X be a Fano variety, i.e. a normal projective complex variety whose anticanonical divisor is Cartier and ample. While Fano varieties with mild singularities have been studied by many authors, we study Fano varieties whose non-klt locus Nklt(X) is not empty. The main result that we will present is the following: if Nklt(X) is not empty, then dim Nklt(X) ≥ k−1, k being the index of X, with equality if and only if Nklt(X) is a linear P^k−1. In this last case, X has lc singularities and is a generalized cone with Nklt(X) as vertex.

Rita Pardini, University of Pisa, Italy

Title: Higher dimensional Clifford-Severi equalities

Abstract: Let X be a projective variety, let a:X-->A be a morphism to an abelian variety with dim a(X)=dim X.Given a line bundle L on X, we define its continuous rank h^0_a(L) as the generic value of h^0(L+P), where P is the pullback of an element of Pic^0(A). The slope of \lambda(L) is the ratio vol(L)/h^0_a(L); the Clifford-Severi inequalities are lower bounds for the slope that generalize at the same time the classical Severi inequality for surfaces of general type and Clifford's theorem for line bundles on a curve. I will describe a characterization of:

(a) the triples (X,a,L) attaining the lower bound \lambda(L)=n! (1st Clifford-Severi line)

(b) the triples (X,a,L) with K_X-L pseff attaining the lower bound \lambda(L)=2n! (2nd Clifford-Severi line).

This is joint work with M.A. Barja and L. Stoppino.

Lorenzo Robbiano, Univerisity of Genova, Italy PDF

Title: Gorenstein, Complete Intersections, Cayley-Bacharach: Algebra, Geometry, and Computer Algebra

Abstract: An affine zero-dimensional K-algebra is a ring of type P/I where K is a field, I is an ideal in P = K[x_1, . . . , x_n], and dimK(P/I) < ∞. In the first part of the talk some features of zero-dimensional affine K-algebras are investigated: having the Cayley-Bacharach property, being locally Gorenstein, and being locally a complete intersection. In particular, the history of these properties and the modern approach via computational methods are discussed. In the second part we enter the realm of border basis schemes. They are open subschemes of the Hilbert scheme of zero-dimensional ideals with a given multiplicity. In particular, we discuss the problem of computing the loci of the above mentioned properties inside the border basis schemes.

Francesco Russo, University of Catania, Italy

Title: Explicit rationality of cubic fourfolds

Abstract: After reviewing the main conjectures and know results about the rationality of cubic fourfolds, we shall present some methods, ideas, results and explicit examples to illustrate the complexity of the problem from different points of view: birational, projective and computational. This is joint work with Giovanni Staglianò.

Andrew Sommese, University of Notre Dame, USA

Title: Numerical Algebraic Geometry and Nonlinear Differential Equations

Abstract: Discretization of many systems of nonlinear differential equations leads to systems of polynomial equations with thousands of variables and equations. For a given system of differential equations, there are many related polynomial systems. E.g., using increasing grid sizes (or different grids) in finite difference methods leads to different systems of polynomials. In recent years, applying numerical-algebraic-geometry approaches to the polynomial systems associated systems of differential equations has led to the numerical solution of the systems of differential equations. For some types of problems, the new solution methods are significantly faster than previous methods. In this talk, I will discuss the polynomial systems arising from discretization, the new methods, and some issues that have arisen in their use.

J. Rafael Sendra, Universidad de Alcala, Spain

Title: Are Rational Parametrizations of Curves and Surfaces Really Applicable?

Abstract: Many of the problems concerning computer aided design and computer aided modeling (CAD/CAM) systems are related to the manipulation of geometric objects, mainly curves, surfaces and their combinations, in two or three-dimensional space. This phenomenon has motivated a reciprocal relationship of interest between the fields of applications and development of constructive methods in algebraic geometry. As a consequence, the development of, both symbolic and/or hybrid symbolic-numeric, algorithms for dealing with curves and surfaces has turned to be an active research area. In this talk we plan to give a panoramic overview of the state of the art of the field. We will describe the main ideas of some of the existing approaches, and we will try to motivate and describe the new challenges in the field.

Edoardo Sernesi, Univerisity of Roma 3, Italy

Title: Gaussian maps on singular K3 curves

Abstract: I will present two results about gaussian maps on the normalization of a 1-nodal curve lying on a K3 surface (one of them is joint work with C. Fontanari).

Thomas Szemberg, University of Cracow, Poland PDF

Title: Unexpected hypersurfaces

Abstract: In a ground-breaking paper of Cook II, Harbourne, Migliore and Nagel the authors introduced a new notion of sets of points admitting unexpected curves. Roughly speaking, they are interested in linear systems of plane curves with assigned base points (a very classical topic in algebraic geometry) and ask when such a system admits members which are singular in a general (hence any) point, even if their existence does not follow from a naive dimension count. I will report on exciting developments in this new field of research. I will also establish some parallels to the theory of higher order embeddings which Mauro Beltrametti co-developed about 30 years ago.

Jaroslaw Wisniewski, University of Warszawa, Poland PDF

Title: Algebraic torus action and contact manifolds

Abstract: Algebraic torus action on projective manifolds provides combinatorial data which may be useful for understanding them. In a joint paper with Jaroslaw Buczynski and Andrzej Weber we use the combinatorics associates to the torus action to prove that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This settles LeBrun-Salamon conjecture about quaternion-Kaehler manifolds in real dimensions 12 and 16.

Francesco Zucconi, University of Udine, Italy

Title: Hyperelliptic spin curves and Fano Vaieties

Abstract: In this talk I'ill review the joint program with Hiromici Takagi to understand some moduli spaces of spin curves by the study of rational curves on some Fano varieties; Special attention will be devoted to the cases of hyperelliptic spin curves.