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# Geometric Methods in Representation Theory Seminar – Philip Tosteson (UNC)

## October 4 @ 4:00 pm - 5:00 pm

**Homology of spaces of curves on blowups**

**Abstract:** Let C be a smooth projective curve and X be a smooth projective variety. We will consider the space degree d algebraic (or holomorphic) maps from C to X.

When X is a projective space, Segal discovered an interesting phenomenon: as the degree increases, the homology of the space of algebraic maps approximates that of the space of continuous maps. Recently, Ellenberg-Venkatesh observed that this phenomenon is related to Manin’s conjectures about rational points on Fano varieties, suggesting it holds more generally.

I will talk about joint work with Ronno Das considering the case where X is a blowup of a projective space at finitely many points (in particular the case of del Pezzo surfaces).