Brill–Noether Theory for (toric) Surfaces and Complete Quasimaps to Blow-ups of Projective spaces

The classical Brill–Noether theorem states that every nondegenerate degree d map from a general curve C of genus g to projective space is a point of expected dimension in the moduli space of such maps. In this talk, I will present an analogous statement for maps from C to smooth projective toric surfaces. I will then discuss the construction of the space of complete quasimaps to Bl_{P^s}^r, obtained as a suitable blow-up of the quasimap space of Ciocan-Fontanine–Kim–Maulik. This space provides an expected-dimension compactification of the moduli space of maps, in a fixed curve class, from C to X. Conjecturally, the insertion of tautological subschemes corresponding to geometric insertions is transverse, lies in the locus of nondegenerate maps, and preserves the expected dimension. Using the Brill–Noether result for toric surfaces mentioned above, the conjecture is verified in dimension 2.