Antonio Lanteri

AMPLE VECTOR BUNDLES WITH SECTIONS VANISHING ON PROJECTIVE SPACES OR QUADRICS

Let ɛ be an ample vector bundle of rank r≥2 on a compact complex manifold X of dimension n≥r+1 having a section whose zero locus is a submanifold Z of the expected dimension n–r. Pairs (X, ɛ) as above are classified under the assumption that Z is either a projective space or a quadric.

The Chow motive of the Godeaux surface

Let X be the Godeaux surface obtained as a quotient of the Fermat quintic in P3C under the appropriate action of Z/5. We show that its Chow motive h(X) splits as 1 ⊕ 9L ⊕ L where L = (P, [P × pt]) is the Lefschetz motive. This provides a purely motivic proof of the Bloch conjecture for X. Our results also give a motivic proof of the Bloch conjecture for those surfaces considered in [BKL], i.e. all surfaces with pg = 0 which are not of general type.

Projective manifolds whose topology is strongly reflected in their hyperplane sections

Let ℒk be the class of complex algebraic k-folds X ⊂ ℙn such that Hi(X, ℚ) ≅ Hi(H, ℚ) for i≤k−1, H a general hyperplane section. Topological characterizations of -k and several other classes of projective manifolds are given. Moreover, classes ℒ2, ℒ3, ℒ5 are completely described and a partial description of ℒ4 is given. A key role is played by projective bundles.

On the 2 and the 3-connectedness of ample divisors on a surface

Let X be a nonsingular complex projective surface and let D be an ample divisor on X such that the associated invertible sheaf is spanned by its global sections. We prove that D is 2-connected apart from a few cases we explicitly describe. We also provide a corresponding result for the 3-connectedness when D2⩾10 and for the 4-connectedness when D2⩾17 and D is very ample.

Algebraic Geometry: A Volume in Memory of Paolo Francia

Preface - Lucian Baxdescu and Michael Schneider: Formal functions, connectivity and homogeneous spaces - Luca Barbieri-Viale: On algebraic 1-motives related to Hodge cycles - Arnaud Beauville: The Szpiro inequality for higher genus fibrations - Giuseppe Borrelli: On regular surfaces of general type with pg=2 and non-birational bicanonical map - Fabricio Catanese and Frank-Olaf Schreyer: Canonical projections of irregular algebraic surfaces - Ciro Ciliberto and Margarida Mendes Lopes: On surfaces with pg=2, q=1 and non-birational bicanonical map - Alberto Conte, Marina Marchisio and Jacob Murre: On unirationality of double covers of fixed degree and large dimension a method of Ciliberto - Alessio Corti and Miles Reid: Weighted Grassmannians - Tommaso de Fernex and Lawrence Ein: Resolution of indeterminacy of pairs - Vladimir Guletskiix and Claudio Pedrini: The Chow motive of the Godeaux surface - Yujiro Kawamata: Francias flip and derived categories - Kazuhiro Konno: On the quadric huil of a canonical surface - Adrian Langer: A note on Bogomolovs instability and Higgs sheaves - Antonio Lanteri and Raquel Mallavibarrena: Jets of antimulticanonical bundles on Del Pezzo surfaces of degree no.2 - Margarida Mendes Lopes and Rita Pardini: A survey on the bicanonical map of surfaces with pg=0 and K2 2 - Francesco Russo: The antibirational involutions of the plane and the classification of real del Pezzo surfaces - Vyacheslav V. Shokurov: Letters of a birationalist IV. Geometry of log flips - Andrew J. Sommese, Jean Verschelde and Charles Wampler A method for tracking singular paths with application to the numerical irreducible decomposition.

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(KX⊗L))/2=3 are classified by means of the main techniques of adjunction theory.

Higher order dual varieties of projective surfaces

We investigate higher order dual varieties of projective manifolds whose osculatory behavior is the best possible. In particular, for a k-jet ample surface we prove the nondegeneratedness of the k-th dual variety and for 2-regular surfaces we investigate the degree of the second dual variety.

Ample Vector Bundles of Curve Genus One

We investigate the pairs (X,E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundleE of rank n− 1 on X such thatE has a section whose zero locus is a smooth elliptic curve.

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Let be an ample vector bundle of rank r = 2 on a smooth complex projective variety X of dimension n such that there exists a global section of whose zero locus Z is a smooth subvariety of dimension n-r = 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, , H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

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.