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Hypersurfaces, moduli of del Pezzos, and Non-reductive GIT

Mathematics Seminar
Thursday, 17 December, 2020 | 14:00–16:30 | Zoom
Contact: Gergely Bérczi

Hypersurfaces, moduli of del Pezzos, and Non-reductive GIT Aarhus Zoom Afternoon 17 December 2020

A zoom meeting to discuss recent progress on moduli of toric hypersurfaces and del Pezzos via NRGIT

Dominic Bunnett (TU Berlin)

17 December 2020 (Thursday), 13:00 GMT

Title: Moduli spaces and stability of toric hypersurfaces

Abstract: The moduli spaces of projective hypersurfaces form a large class of moduli spaces of varieties and are constructed as quotient spaces via reductive GIT. Recently the GIT-stability of hypersurfaces has appeared again in relation to K-stability and the moduli of K-stable Fano varieties. We review the moduli and GIT-stability of hypersurfaces, taking a tour of what's known and some open questions. We shall then generalise the moduli problem to hypersurfaces in toric varieties, where the moduli spaces constructed in these cases play a significant role in Mirror symmetry. We prove that in the case of weighted projective space, using Non-reductive GIT, these moduli spaces are DM stacks with quasi-projective coarse moduli spaces and in certains low degree and dimensional cases, we can completely describe their geometry.

Joshua Jackson (Imperial)

17 December 2020 (Thursday), 14:15 GMT

Title: Moduli of del Pezzo surfaces via NRGIT

Abstract: Following on from the previous talk, I will outline the basic theory of Non-reductive GIT due to Berczi-Doran-Hawes-Kirwan, and explain in more detail how it is applied in the case of moduli of hypersurfaces. I will then report on joint work-in-progress with the other speaker, which uses these methods to construct moduli spaces of del Pezzo surfaces of degree one and two. We proceed after the fashion of the constructions of Odaka-Spotti-Sun via reductive GIT, which were motivated by Gromov-Hausdorff compactification questions coming from K-stability. I will show that the moduli space constructed via NRGIT coincides with that of Odaka-Spotti-Sun in the degree two case, and in the degree one case gives a potentially different compactification of the set of smooth del Pezzos.

Zoom link

https://aarhusuniversity.zoom.us/j/9494116175?pwd=VlJWMERJc3N3UnBGSWJTL3hIbUJGZz09

Meeting ID: 949 411 6175

Passcode: Aarhus