# Graduate Studies A: Elliptic Surfaces - Classical to Moduli. Spring 2023.

**Instructor:** June Park (june[dot]park [at] unimelb [dot] edu [dot] au)

**Time and place:** Tuesday 3:15-4:15pm + Fridays 10:00-11:00am in Russel Love Theatre, Peter Hall Building.

**Office hours:** By appointment, G82 Peter Hall Building.

## Syllabus

### Course description

The study of elliptic fibrations f : X → C of an elliptic surface X over a curve C (i.e. family of elliptic curves over a global field) lies at the heart of surface classification.John Tate (1925-2019) gave us a wealth of essential mathematical ideas and constructions in algebraic number theory and arithmetic geometry.

In this class, I will unpack the classical Tate's algorithm by explicitly working with polynomials and drawing the procedure by the help of the theory of moduli stacks.

### Overview of lectures

1. Organizational remarks, Motivation, Review on elliptic curves and Riemann-Roch for projective curves
- 02/21/2023.

Zoom recording of Lecture 1 (Lecture starts about ten minutes into the video) Passcode: W&HzTu2J

2. Intersection forms on smooth 4-manifolds, Rational & Elliptic K3 surfaces, Topological classification - 02/24/2023.

Zoom recording of Lecture 2 Passcode: k319.DqR

3. Elliptic fibrations via Lefschetz pencils, Fiber sum, Mapping class groups with monodromy factorization - 02/28/2023.

Zoom recording of Lecture 3 Passcode: UJM6B%ea

4. Relationship between smooth 4-manifolds and algebraic surfaces, Enriques–Kodaira classification, Geography - 03/3/2023.

5. Elliptic fibrations via Weierstrass models, Fundamental line bundle, Discriminants, Reduction of elliptic curves, Stable reduction, Tate’s Algorithm - 03/7/2023.

6. Birational geometry of surfaces, Minimal models and resolutions, Kodaira's classification of singular fibres for minimal elliptic surfaces - Lecture by Dougal Davis - 03/10/2023.

Zoom recording of Lecture 6 Passcode: x?ujda8k

7. Fine moduli (space) of elliptic curves, Universal family, Automorphisms - 03/14/2023.

Zoom recording of Lecture 7 Passcode: LC@83ip5

8. More on algebraic stacks, Stable curves in genus 1, Moduli stacks of curves, Deligne-Mumford compactification - 03/17/2023.

Zoom recording of Lecture 8 Passcode: b^y^oS1h

9. Fibrations of algebraic curves, Hom stacks of curves, DEF=DIFF of elliptic surfaces - 03/21/2023.

Zoom recording of Lecture 9 Passcode: C3#c=6DU

I had trouble starting Zoom due to the software update. I suggest that you use the blackboard captured at 00:36:31 & 00:15:59 to make sense of the lecture.

10. Crash course in Arithmetic Topology : Grothendieck motivic classes / Point counts over finite fields, Étale cohomology with Weil conjecture - 03/24/2023.

11. Stratification of the moduli, Moduli space of coprime polynomials and morphisms, Work of Segal, Farb and Wolfson - 03/28/2023.

12. Motives and Cohomology of Hom stacks and Enumerating elliptic curves over function fields - 03/31/2023.

13. Integral / Rational points on M_{1,1} over global fields, Heights on Stacks, Tate's algorithm via twisted maps - 04/4/2023.

### References

#### Books, papers and notes:

From the __4-Manifolds__ perspective:

Scorpan, *The Wild World of 4-Manifolds.* American Mathematical Society, 2005.

Gompf–Stipsicz, *4-Manifolds and Kirby Calculus.* American Mathematical Society, 1999.

Kirby, *The Topology of 4-Manifolds.* Springer, 1989.

From the __Algebraic Surfaces__ perspective:

Barth–Hulek–Peters–Van de Ven, *Compact complex surfaces.* Springer, 2004.

Schütt–Shioda, *Elliptic surfaces.* Mathematical Society of Japan, 2010.

Miranda, *The basic theory of elliptic surfaces.* Pisa, 1989.

From the __Arithmetic Topology__ perspective:

Han–Park, *Arithmetic of the moduli of semistable elliptic surfaces.* Mathematische Annalen, 2019.

Farb–Wolfson, *Topology and arithmetic of resultants, I.* New York Journal of Mathematics, 2016.

Björner–Ekedahl, *Subspace arrangements over finite fields: cohomological and enumerative aspects.* Advances in Mathematics, 1997.

Segal, *The topology of spaces of rational functions.* Acta Mathematica, 1979.

Hell, if you made it this far, it's about time to start reading the __Prerequisites__:

Alper, *Stacks and Moduli.* Notes, 2023.

Liu, *Algebraic geometry and arithmetic curves.* Oxford, 2002.

Mumford, *The red book of varieties and schemes.* Springer, 1999.