On the singular sheaves in the fine Simpson moduli spaces of 1-dimensional sheaves

in Canadian Mathematical Bulletin (2016)

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M'$ of sheaves that are not locally free on their support. We ... [more ▼]

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M'$ of sheaves that are not locally free on their support. We show that for $d\ge 4$ it is a singular subvariety of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted as a (partial) modification of $M\setminus M'$ by line bundles (on support). [less ▲]

On 1-dimensional sheaves on projective plane

Presentation (2016, January)

Let M be the Simpson moduli space of semistable sheaves on the projective plane with fixed linear Hilbert polynomial P(m)=dm+c. A generic sheaf in M is a vector bundle on its Fitting support, which is a ... [more ▼]

Let M be the Simpson moduli space of semistable sheaves on the projective plane with fixed linear Hilbert polynomial P(m)=dm+c. A generic sheaf in M is a vector bundle on its Fitting support, which is a planar projective curve of degree d. The sheaves that are not vector bundles on their support constitute a closed subvariety M' in M. We study the geometry of M' in the case of Hilbert polynomials dm-1 (for d bigger than 3) and demonstrate that M' is a singular variety of codimension 2 in M. We speculate on how the question we study is related to recompactifying of the Simpson moduli spaces by vector bundles. [less ▲]

A global description of the fine Simpson moduli space of 1-dimensional sheaves supported on plane quartics

E-print/Working paper (2016)

A global description of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican and ... [more ▼]

A global description of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican and show that the Simpson moduli space M=M_{4m±1}(P_2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. An easy computation of the Poincaré polynomial of M is presented. [less ▲]

Riemann Surfaces. Lecture notes. Winter semester 2015/2016.

Learning material (2015)

These are the lecture notes for the course 'Riemann surfaces' (14 Lectures, 90 minutes each). The lectures are provided with exercises.

On the singular sheaves in the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics

in Rendiconti dell'Istituto di Matematica dell'Università di Trieste (2015)

In the case of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics, the subvariety of sheaves that are not locally free on their support is connected, singular, and has ... [more ▼]

In the case of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics, the subvariety of sheaves that are not locally free on their support is connected, singular, and has codimension 2. [less ▲]

lrcalc.lib: a Singular interface to the Littlewood-Richardson Calculator by Anders Buch

Software (2015)

This library for the computer algebra system Singular is an interface for the Littlewood-Richardson Calculator by Anders Buch.

Universal plane curve and moduli spaces of 1-dimensional coherent sheaves

E-print/Working paper (2014)

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable ... [more ▼]

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable locus. The universal singular locus coincides with the subvariety of M consisting of sheaves that are not locally free on their support. It turns out that the blow up of M along M' may be naturally seen as a compactification of M_B = M\M' by vector bundles (on support). [less ▲]

Riemann Surfaces. Lecture notes. Winter semester 2013/2014.

Learning material (2013)

This is a preliminary version of lecture notes for the course 'Riemann surfaces' (14 Lectures, 90 minutes each).

Universal plane curves and Simpson moduli spaces of 1-dimensional sheaves

Poster (2012, June)

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable ... [more ▼]

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable locus. The universal singular locus coincides with the subvariety of M consisting of sheaves that are not locally free on their support. It turns out that the blow up of M along M' may be naturally seen as a compactification of M_B = M\M' by vector bundles (on support). [less ▲]