1,721,045 research outputs found
Lectures on Brill-Noether theory
No full textThese notes are the summary of lectures given by the author, in the framework of Joint Lectures of F. Flamini and E. Sernesi, at the Workshop
”Curves and Jacobians”, organized by the Korean Institute of Advanced Study (Seoul) and held on October 18-21, 2010, at Sol Beach Resort, Yangyang (Korea
Equivalence of families of singular schemes on threefolds and fourfolds
No full textThe main purpose of this paper is twofold. We first
analyze in detail the meaningful geometric aspect of the method
introduced in previous papers of the author, concerning
families of irreducible, nodal "curves" on a smooth, projective threefold X.
This analysis gives some geometric interpretations not investigated in the previous papers and
highlights several interesting connections with families of other singular geometric
"objects" related to X and to other varieties.
Then, we use this method to study analogous
problems for families of singular divisors on ruled fourfolds suitably
related to X. This enables us to show that Severi varieties of vector bundles
on X can be rephrased in terms of "classical" Severi varieties of divisors
on such fourfolds
A first course in Algebraic Geometry and Algebraic Varieties
No full textThis book provides a gentle introduction to the foundations of Algebraic Geometry, starting from computational topics (ideals and homogeneous ideals, zero loci of ideals) up to increasingly intrinsic and abstract arguments, such as "Algebraic Varieties", whose natural continuation is a more advanced course on the theory of schemes, vector bundles, and sheaf-cohomology.
Valuable to students studying Algebraic Geometry and Geometry, this title contains around 60 exercises (with solutions) to help students thoroughly understand the theories introduced in the book. Proofs of the results are carried out in full detail. Many examples are discussed in order to reinforce the understanding of both the theoretical elements and their consequences, as well as the possible applications of the material
Some results of regularity for Severi varieties of projective surfaces
No full textFor a linear system |C|
on a smooth projective surface
S, whose general member is a smooth, irreducible
curve, the Severi variety V_{|C|, d} is the
locally closed subscheme of |C| which parametrizes curves
with only d
nodes as singularities. In this paper we give numerical conditions
on the class of divisors and upper bounds on d, ensuring
that the corresponding Severi variety is smooth of codimension d.
Our result generalizes what is proven in previous papers in the literature.
We also consider examples of smooth Severi varieties on surfaces of general
type in P^3 which contain a line
IP^r scrolls arising from Brill-Noether theory and K3-surfaces
No full textIn this paper we study examples of P(r)-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from some results of Lazarsfeld. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of F(g), the moduli space of such K3' s, and M(v)(S), the moduli space of semistable torsion-free sheaves of a given rank on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a very general K3 as well as Brill-Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in Calabri et al. (Rend LinceiMat Appl 17(2): 95-123, 2006) and Calabri et al. (Rend Circ Mat Palermo 57(1): 1-32, 2008)
Ulrich bundles on some threefold scrolls over F_e
Full text linkWe investigate the existence of Ulrich vector bundles on suitable 3-fold scrolls Xe over Hirzebruch surfaces Fe, for any e⩾0, which arise as tautological embedding of projectivization of very-ample vector bundles on Fe which are uniform in the sense of Brosius and Aprodu--Brinzanescu (cf. [8] and [3], respectively, in Bibliography). We explicitely describe components of moduli spaces of rank r⩾1 Ulrich vector bundles whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components as well as we prove that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such Xe and give an effective proof of the fact that such Xe's turn out to be geometrically Ulrich wild
Big Vector Bundles on Surfaces and Fourfolds
Full text linkThe aim of this note is to exhibit explicit sufficient cohomological criteria ensuring bigness of globally generated, rank-r vector bundles, r⩾ 2 , on smooth, projective varieties of even dimension d⩽ 4. We also discuss connections of our general criteria to some recent results of other authors, as well as applications to tangent bundles of Fano varieties, to suitable Lazarsfeld–Mukai bundles on fourfolds, etcetera
Moduli spaces of bundles and of Hilbert schemes over -gonal curve
No full textThe aim of this paper is two--fold. We first strongly improve our previous main result published in Proc. Math. Am. Soc., concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees , with at least independent global sections, on a general --gonal curve of genus .
We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable
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