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Search Results: 1 - 10 of 1288 matches for " Marcello Bernardara "
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Calabi-Yau complete intersections with infinitely many lines
Marcello Bernardara
Mathematics , 2004,
Abstract: We give two new examples of families of Calabi-Yau complete intersection threefolds whose generic element contains infinitely many lines. We get some results about the normal bundles of these lines and the Hilbert scheme of lines on the threefolds.
A semiorthogonal decomposition for Brauer Severi schemes
Marcello Bernardara
Mathematics , 2005,
Abstract: A semiorthogonal decomposition for the bounded derived category (the category of perfect complexes in a non smooth case) of coherent sheaves on a Brauer Severi scheme is given. It relies on bounded derived categories (categories of perfect complexes in a non smooth case) of suitably twisted coherent sheaves on the base.
Stable pairs on elliptic K3 surfaces
Marcello Bernardara
Mathematics , 2008,
Abstract: We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.
Fourier-Mukai transforms of curves and principal polarizations
Marcello Bernardara
Mathematics , 2007,
Abstract: Given a Fourier-Mukai transform $\Phi$ between the bounded derived categories of two smooth projective curves, we verifiy that the induced map between the Jacobian varieties preserves the principal polarization if and only if $\Phi$ is an equivalence.
Derived categories and rationality of conic bundles
Marcello Bernardara,Michele Bolognesi
Mathematics , 2010,
Abstract: We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.
Categorical representability and intermediate Jacobians of Fano threefolds
Michele Bolognesi,Marcello Bernardara
Mathematics , 2011,
Abstract: We define, basing upon semiorthogonal decompositions of $\Db(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.
Relations between the Chow motive and the noncommutative motive of a smooth projective variety
Marcello Bernardara,Goncalo Tabuada
Mathematics , 2013,
Abstract: In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Examples, counter-examples, and applications are also described.
Additive invariants of finite dimensional algebras of finite global dimension
Marcello Bernardara,Goncalo Tabuada
Mathematics , 2013,
Abstract: Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value of every additive invariant at A only depends on the number of simple modules. Examples of additive invariants include algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Along the way we establish two results of general interest. The first one concerns the compatibility between bilinear pairings and Galois actions on the Grothendieck group of every proper dg category. The second one is a transfer result in the setting of noncommutative motives.
Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
Asher Auel,Marcello Bernardara
Mathematics , 2015,
Abstract: We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles.
The Euclid-Fourier-Mukai algorithm for elliptic surfaces
Marcello Bernardara,Georg Hein
Mathematics , 2010,
Abstract: We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.
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