Irreducibility of the moduli space of stable vector bundles of rank two and odd degree on a very general quintic surface

Abstract

The moduli space M (c 2 ), of stable rank two vector bundles of degree one on a very general quintic surface X ⊂ P 3 , is irreducible for all c 2 ≥ 4 and empty otherwise.

Geometry of certain Brill-Noether locus on a very general sextic surface and Ulrich bundles

“…Recently in [34], it's shown that M(2; H, c 2 ) is good (i.e generically smooth of the expected dimension) for c 2 ≥ 20 and irreducible for c 2 ≥ 27. Also from [32] it's known that M(2; H, 11) is reducible with atleast 2 irreducible components (which is the first example of its kind). In a recent work of the present author with S.Pal, it's established that M(2; H, c 2 ) is irreducible for c 2 ≤ 10 with the space being non-empty for c 2 ≥ 5, c 2 = 7 (cf.…”

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson Conjecture

“…If, for all k ≥ 1, no dk + 2 points of P lie in a projective k−plane, then P impose independent conditions on forms of degree d; in fact there is a multilinear form of degree d containing any subset consisting of all but one of the points, but missing the last. (r, d, n) is one of the following: (2,4,5), (3,4,9), (4,4,14), (4,3,7).…”

“…When the underlying surface S is a very general quintic hypersurface in P 3 Simpson and Mestrano studied this question systematically and in [10], the current author with K. Dan, studied the question related to Brill-Noether loci. In a series of papers [1], [2], [3], Simpson and Mestrano proved that the moduli space of rank 2 H−stable torsion free sheaves with fixed determinant H is generically smooth and irreducible, where H := O S (1). This result was known before by an unpublished work by Nijsse for c 2 ≥ 16 [22].…”

“…In [2], Simpson and Mestrano showed that the moduli space of stable rank 2 bundles over sextic surface is not irreducible for c 2 = 11 . In fact, they have shown that the moduli space in this case has at least two different irreducible components.…”

On the Bogomolov-Gieseker inequality for hypersurfaces in the projective spaces

“…Theorem 1.2 for higher rank bundles is completely new. See also [MS11,MS18,Tod14] for other results concerning rank two vector bundles.…”