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In these lectures, I will discuss recent developments in the birational
geometry of moduli spaces of sheaves on surfaces motivated by
Bridgeland stability conditions. My emphasis will
be on concrete examples. After reviewing the basic facts about Hilbert
schemes of points and moduli spaces of vector bundles on surfaces, I
will introduce Bridgeland stability
conditions. I will then describe joint work with Arcara, Bertram and
Huizenga on the birational geometry of Hilbert schemes of points on the
projective plane. Building from there, I
will discuss recent joint work with Huizenga on computing nef cones of
moduli spaces of sheaves on surfaces, often using the projective plane
as the motivating example.
SHORT BIBLIOGRAPHY for the mini-course: (1) Elementary lectures for beginning students: http://homepages.math.uic.edu/~coskun/CIMPA.pdf (2) Lecture notes giving an overview of the projective plane case: http://homepages.math.uic.edu/~coskun/gokova.pdf (3) The case of the projective plane: http://homepages.math.uic.edu/~coskun/hilbbridge-1.pdf http://homepages.math.uic.edu/~coskun/conesOfBundles.pdf http://homepages.math.uic.edu/~coskun/amplecone.pdf (4) The case of K3 surfaces: http://arxiv.org/pdf/1203.4613.pdf http://arxiv.org/pdf/1301.6968.pdf (5) The case of an arbitrary surface: http://homepages.math.uic.edu/~coskun/amplecone.pdf |