Search

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring. [ABSTRACT FROM AUTHOR]

The moduli space of rank 2 stable, locally-free sheaves with fixed odd degree determinant over a smooth, projective curve is a classical object. In the early 1970s, Newstead gave the generators of the cohomology ring of this moduli space. There are generalizations of this result to higher rank, but nothing is known for the case when the underlying curve is singular. In this article, we generalize Newstead's result to the case when the underlying curve is irreducible, nodal. We show that the generators of the cohomology ring in the nodal curve case arise naturally as degeneration of Newstead's generators in the smooth curve case. [ABSTRACT FROM AUTHOR]

For d ≥ 4 , the Noether–Lefschetz locus NL d parametrizes smooth, degree d surfaces in ℙ 3 with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the Noether–Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large d , but is true for d ≤ 5. She also showed that for d = 6 , 7 , there are finitely many reduced, irreducible components of NL d of non-maximal codimension. In this paper, we prove that for any d ≥ 6 , there are infinitely many non-reduced irreducible components of NL d of non-maximal codimension. [ABSTRACT FROM AUTHOR]

Given a smooth, projective variety X and an effective divisor D ⊆ X , it is well-known that the (topological) obstruction to the deformation of the fundamental class of D as a Hodge class, lies in H 2 ( O X) . In this article, we replace H 2 ( O X) by H D 2 ( O X) and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of D as an effective Cartier divisor of a first order infinitesimal deformations of X). We apply this to study the jumping locus of families of linear systems and the Noether–Lefschetz locus. Finally, we give examples of first order deformations X t of X for which the cohomology class [D] deforms as a Hodge class but Ddoes not lift as an effective Cartier divisor of X t . [ABSTRACT FROM AUTHOR]

Let π 1 : X → Δ be a flat family of smooth, projective curves of genus g ≥ 2 , degenerating to an irreducible nodal curve X 0 with exactly one node. Fix an invertible sheaf L on X of relative odd degree. Let π 2 : G (2 , L) → Δ be the relative Gieseker moduli space of rank 2 semi-stable vector bundles with determinant L over X . Since π 2 is smooth over Δ ∗ , there exists a canonical family ρ ~ i : J G (2 , L) Δ ∗ i → Δ ∗ of i-th intermediate Jacobians i.e., for all t ∈ Δ ∗ , (ρ ~ i) - 1 (t) is the i-th intermediate Jacobian of π 2 - 1 (t) . There exist different Néron models ρ ¯ i : J ¯ G (2 , L) i → Δ extending ρ ~ i to the entire disc Δ , constructed by Clemens, Saito, Schnell, Zucker and Green–Griffiths–Kerr. In this article, we prove that in our setup, the Néron model ρ ¯ i is canonical in the sense that the different Néron models coincide and is an analytic fiber space which graphs admissible normal functions. We also show that for 1 ≤ i ≤ max { 2 , g - 1 } , the central fiber of ρ ¯ i is a fibration over product of copies of J k (Jac (X ~ 0)) for certain values of k, where X ~ 0 is the normalization of X 0 . In particular, for g ≥ 5 and i = 2 , 3 , 4 , the central fiber of ρ ¯ i is a semi-abelian variety. Furthermore, we prove that the i-th generalized intermediate Jacobian of the (singular) central fibre of π 2 is a fibration over the central fibre of the Néron model J ¯ G (2 , L) i . In fact, for i = 2 the fibration is an isomorphism. [ABSTRACT FROM AUTHOR]

A theorem of Weil and Atiyah says that a holomorphic vector bundle E on a compact Riemann surface X admits a holomorphic connection if and only if the degree of every direct summand of E is zero. Fix a finite subset S of X, and fix an endomorphism $$A(x) \in \text {End}(E_x)$$ for every $$x \in S$$ . It is natural to ask when there is a logarithmic connection on E singular over S with residue A( x) at every $$x \in S$$ . We give a necessary and sufficient condition for it under the assumption that the residues A( x) are rigid. [ABSTRACT FROM AUTHOR]

A classical example of Mumford gives a generically non-reduced component of the Hilbert scheme of smooth curves in [ABSTRACT FROM AUTHOR]

Let be a holomorphic principal -bundle on a compact connected Riemann surface , where is a connected reductive complex affine algebraic group. Fix a finite subset , and for each fix . Let be a maximal torus in the group of all holomorphic automorphisms of . We give a necessary and sufficient condition for the existence of a -invariant logarithmic connection on singular over such that the residue over each is . We also give a necessary and sufficient condition for the existence of a logarithmic connection on singular over such that the residue over each is , under the assumption that each is -rigid. [ABSTRACT FROM AUTHOR]

In this article, we study subloci of solvable curves in Mg which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus g curves in a cubic surface, we show that a general such curve is solvable. [ABSTRACT FROM AUTHOR]

For any integer d ≥ 5, the Noether-Lefschetz locus, denoted d, parametrizes smooth degree d surfaces in ℙ3 with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of d is of codimension (in the space of all smooth surfaces in ℙ3 of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of d, denoted r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of r,d is of codimension and the irreducible component of r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d. [ABSTRACT FROM AUTHOR]