Informal AG Seminar (Fall 2022)

Organizers: Andres Fernandez Herrero, Nathan Chen

Topic: Vector bundle methods and Brill Noether theory

Time: 4:30pm ET on Tuesdays, Fall 2022

Location: Room 520 (unless otherwise noted), Columbia math department

Here is a tentative schedule of the talks (these will be updated as we go along):

• September 20 (Nathan Chen): Ample + nef: definitions, properties, examples. See 1.2-1.5 of [Positivity1] [notes]

• September 27 (Amal Mattoo): Castelnuovo-Mumford regularity. See 1.8.A and parts of 1.8.B of [Positivity1]

• October 4 (Gyujin Oh): Foundations of Brill Noether theory and Petri's condition. See Chapter V of [ACGH] and 3.1 of [VBtechniques]

• October 11 (Matthew Hase-Liu): Brill-Noether theory on K3 surfaces and Lazarsfeld-Mukai bundles. See 3.2 and 3.3 of [VBtechniques]

• October 18 (Benjamin Church): Mukai's theorem and proof of Brill-Noether-Petri. See 3.3 of [VBtechniques], first half of 3.4 -- stop at end of first proof of Theorem 3.2.1.(i).

• October 25 (Simon Felten): Reider's theorem and stable bundles. See 4.1 and 4.2 of [VBtechniques]

• November 1 (Andres Fernandez Herrero): Miyaoka's proof of Bogomolov's theorem. See 4.2 and 4.3 of [VBtechniques]

• November 8 (Nicolas Vilches Reyes): Proof of Reider's theorem and applications. See 4.4 of [VBtechniques]

• November 15 (Nathan Chen): Presenting a paper of Izzet Coskun, Eric Larson, Isabel Vogt: "The normal bundle of a general canonical curve of genus at least 7 is semistable".

• November 22: Thanksgiving.

• November 29 (Andres Fernandez Herrero): modern stability results of Langer.

References:

• [ACGH] -- Geometry of algebraic curves volume 1 by Arbarello, Cornalba, Griffiths, Harris

• Brill-Noether-Petri without degenerations. See https://www.math.stonybrook.edu/robert.lazarsfeld/Reprints/Lazarsfeld.BNP.pdf

• [Positivity1] -- Positivity in algebraic geometry volume 1 by Lazarsfeld

• [VBtechniques] -- A sampling of vector bundle techniques in the study of linear series by Lazarsfeld