Antony Maciocia

Complex surfaces with equivalent derived categories

Abstract. We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves (DX). To do this we find, for each such surface X, the set of surfaces Y for which there exists a Fourier-Mukai transform D

Fourier-Mukai transforms for K3 and elliptic fibrations

(Y)\to

Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces

D(X).

Rank 1 Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface

Given a non-singular variety with a K3 fibration π : X → S we construct dual fibrations π : Y → S by replacing each fibre Xs of π by a two-dimensional moduli space of stable sheaves on Xs. In certain cases we prove that the resulting scheme Y is a non-singular variety and construct an equivalence of derived categories of coherent sheaves Φ: D(Y ) → D(X). Our methods also apply to elliptic and abelian surface fibrations. As an application we use the equivalences Φ to relate moduli spaces of stable bundles on elliptic threefolds to Hilbert schemes of curves.

Fourier-Mukai transforms and Bridgeland stability conditions on Abelian threefolds II

We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when the number of walls is globally finite. In examples, we show how to use the explicit conditions to locate walls and sometimes to show that there are no walls at all.

Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles

We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface Graphic corresponding to twisted ideal sheaves. We use Fourier–Mukai techniques to extend the ideas of Arcara and Bertram to express wall crossings as Mukai flops and show that the moduli spaces are projective.

Instanton Moduli as a Novel Map from Tori to K3-Surfaces

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.

Critical

By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces

k

X

-very ampleness for abelian surfaces

the punctual Hilbert schemes