In the last five years Bridgeland stability has
revolutionized our understanding of the geometry of moduli spaces
of sheaves on surfaces, allowing us to compute ample and effective
cones and describe different birational models. In this talk, I
will survey some of my joint work with Daniele Arcara, Aaron
Bertram, Jack Huizenga and Matthew Woolf on the birational geometry
of moduli spaces of sheaves on the plane. I will describe the ample
and effective cones of these moduli spaces, concentrating on
Hilbert schemes of points and concrete examples.
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