Boundary of the moduli space of instanton bundles on projective space

• Grant number: 14/22807-6
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: April 02, 2015
• End date: June 26, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Marcos Benevenuto Jardim
• Grantee: Marcos Benevenuto Jardim
• Visiting researcher: Alexander Tikhomirov
• Visiting researcher institution: Laboratory of Algebraic Geometry and its Applications, Russia
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

We study the Gieseker-Maruyama moduli space of semistable rank two coherent sheaves with zero first and third Chern classes and positive second Chern class (charge) on projective space. Our goal is to understand the geography and the geometry of components of this space. The Gieseker-Maruyama space has as an open subset the moduli space of stable rank two vector bundles with trivial determinant and given charge. The last space contains as an irreducible component the moduli space of mathematical instanton vector bundles of given charge. Our second goal is to describe the boundary of the closure of mathematical instantons in the Gieseker-Maruyama space.The first conjecture which we are going to prove is that via elementary transformations of instanton bundles of lower charge along smooth space curves of an arbitrary genus and an appropriate degree we obtain a non-locally free instanton sheaf of a given charge, and such sheaves constitute a component of the Gieseker-Maruyama space. As a corollary of this result it will follow that the number of irreducible components of the Gieseker-Maruyama space the generic points of which are non-locally free sheaves tends to infinity as the charge grows. Our next task is to prove that the intersections of the above mentioned instanton components with the boundary of the closure of the moduli space of mathematical instantons are divisorial in thisclosure, and that these divisors are distinct from divisorial components of the boundary constructed earlier by Jardim, Markushevich and Tikhomirov. We also are going to describe genericpoints of these divisors in terms of theta characteristics on curves of singularities of generic sheaves from these divisors.The last task in order is to study those sheaves in the boundary of the closure of mathematical instantons which have zero-dimensional singularities. We are going to prove that such components of the boundary are divisorial in the closure of instantons. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

JARDIM, MARCOS; MARKUSHEVICH, DIMITRI; TIKHOMIROV, ALEXANDER S.. Two infinite series of moduli spaces of rank 2 sheaves on P-3. Annali di Matematica Pura ed Applicata, v. 196, n. 4, p. 1573-1608, AUG 2017. (14/22807-6, 14/14743-8)