Moduli of roots of line bundles on curves

We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.

     

Moduli space of parabolic vector bundles over hyperelliptic curves

Let X be a smooth projective hyperelliptic curve of arbitrary genus g. In this article, we will classify the rank 2 stable vector bundles with parabolic structure along a reduced divisor of degree 4.     

Moduli for principal bundles over algebraic curves: II

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford’s geometric invariant theory. This is the second and concluding part of the thesis of late Professor A Ramanathan; the first part was published in the previous issue.     

Moduli for principal bundles over algebraic curves: I

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford's geometric invarian theory.     

Moduli spaces of (semi)stable vector bundles on tree-like curves

Partially supported by my husband during the preparation of this work     

Moduli of products of curves

Some technical results on the deformations of varieties of general type and on permanence of semi-log-canonical singularities are proved. These results are applied to show that the connected component of the moduli space of stable surfaces containing the moduli point of a product of stable curves is the product of the moduli spaces of the curves, assuming the curves have different genera. An application of this result shows that even after compactifying the moduli space and fixing numerical invariants, the moduli spaces are still very disconnected.Received: 20 February 2004     

Moduli of vector bundles and numbers of classes

Kuibyshev State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 1, pp. 81–83, January–March, 1991.     

Moduli of twisted spin curves

In this note we give a new, natural construction of a compactification of the stack of smooth -spin curves, which we call the stack of stable twisted -spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible -spaces and -line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves.

We construct representable morphisms from the stacks of stable twisted -spin curves to the stacks of stable -spin curves and show that they are isomorphisms. Many delicate features of -spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the -operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.