Laura Costa

Vector bundles on fano 3-folds without intermediate cohomology

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology.

Nondegenerate multidimensional matrices and instanton bundles

In this paper we prove that the moduli space of rank 2n symplectic instanton bundles on P 2n+1 , defined from the well-known monad condition, is affine. This result was not known even in the case n = 1, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills Sp(1)-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multi-dimensional matrices representing the instanton bundles.

Derived categories of projective bundles

The goal of this short note is to prove that any projective bundle P(e) → X has a tilting bundle whose summands are line bundles whenever the same holds for X.

Rationality of moduli spaces of vector bundles on rational surfaces

Let X be a smooth rational surface. In this paper, we prove the rationality of the moduli space M X,L (2; c 1 ; c 2 ) of rank two L -stable vector bundles E on X with det ( E ) = c 1 ∈ Pic(X) and c 2 (E) = c 2 ≫ 0.

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category Db(X), where X is the blow up of ℙn−r×ℙr along a multilinear subspace ℙn−r−1×ℙr−1 of codimension 2 of ℙn−r×ℙr. As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.

On the rationality of moduli spaces of vector bundles on Fano surfaces

In this paper we prove the rationality of the moduli space of rank two, H-stable vector bundles on Fano surfaces. @ 1999 Elsevier Science B.V. All rights reserved.

Rationality of moduli spaces of vector bundles on Hirzebruch surfaces

In the 1980’s, Donaldson proved that

Brill-Noether theory for moduli spaces of sheaves on algebraic varieties

and ML(2; cl, c2) are generically smooth of the expected dimension provided c2 is large enough ([D]). As a consequence he obtained some spectacular new results on classification of

Group Action on Instanton Bundles over ℙ3

four manifolds. Since then, many mathematicians have studied the structure of the moduli spaces

Ulrich Bundles on Veronese Surfaces

(resp. ML(r; cl, c2)) from the point of view of algebraic geometry, of topology and of differential geometry; giving very pleasant connections between these areas.