Webpage for the Reading seminar on virtual intersection theory

The goal of this reading group would be to:

Review intersection theory on schemes and its extensions to stacks.
Understand how obstruction theories are related to deformation theory in algebraic geometry, with examples from moduli theory.
Construct a virtual fundamental class from a perfect obstruction theory.
Constuct virtual pullbacks, a generalization of virtual fundamental classes.
Prove the virtual localization formula, and do some applications.
Along the way, prove the axioms of Gromov-Witten Theory.

A tentative schedule (up to debate) would be

Lecture	Week	Topic	Lecturer	References
1	16 Sept-22 Sept	Organizational meeting + introduction to the topic	Aitor	[1]
2	23 Sept-30 Sept	Intersection theory: cones and Gysin pullbacks.	tbd	[4], Chapters 4,5,6
3	30 Sept-6 Oct	Intersection theory on stacks	tbd
4	7 Oct-13 Oct	Cone stacks	tbd	[3]
5	14 Oct-20 Oct	Stacks of the form $[h^1/h^0]$	tbd	[3]
6	21 Oct-27 Oct	Deformation theory I: deformations and the cotangent complex	tbd	[3] and [0]
7	28 Oct-3 Nov	Deformation theory II: obstructions and obstruction theories. With examples	tbd	[3] and [0]
8	4 Nov-10 Nov	The intrinsic normal cone and the virtual fundamental class	tbd	[3]
9	11 Nov-17 Nov	Properties of the virtual fundamental class. The axioms of Gromov-Witten theory	tbd	[3] and [2]
10	18 Nov-24 Nov	Virtual pullbacks I: construction	tbd	[6]
11	25 Nov-1 Dec	Virtual pullbacks II: properties	tbd	[6]
12	2 Dec-8 Dec	The virtual localization formula I: definitions and the usual localization formula	tbd	[5]
13	9 Dec-15 Dec	The virtual localization formula II: proof of the main theorem	tbd	[5] and [7]
14	16 Dec-22 Dec	The virtual localization formula III: Applications	tbd	[5], maybe [8]

If you want to read what this is all about before coming, [1] is a very friendly text.

References:

The best reference is probaly

[0] Lecture notes from a seminar that Alessio and Miguel (former students at ETH) held a few years ago.

The rest of the references are:

[1] Virtual fundamental classes for the working mathematician L. Battistella, F. Carocci and C. Manolache (https://arxiv.org/abs/1804.06048)
[2] Gromov-Witten invariants in Algebraic Geometry K. Behrend (https://arxiv.org/abs/alg-geom/9601011)
[3] The intrinsic normal cone K. Behrend and B. Fantechi (https://arxiv.org/abs/alg-geom/9601010)
[4] Intersection Theory W. Fulton (https://link.springer.com/book/10.1007/978-1-4612-1700-8)
[5] Localization of virtual classes T. Graber and R. Pandharipande (https://arxiv.org/abs/alg-geom/9708001)
[6] Virtual pullbacks C. Manolache (https://arxiv.org/abs/0805.2065)
[7] An invitation to modern enumerative geometry A. T. Ricolfi (https://link.springer.com/book/10.1007/978-3-031-11499-1)
[8] Hodge integrals and Hurwitz numbers via localization T. Graber and R. Vakil (https://arxiv.org/abs/math/0003028)