# 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 function 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, we will review the literature on algebraic surfaces and 4-manifolds, unpack the classical Tate's algorithm by explicitly working with polynomials

and drawing the procedure by the help of recent theory of heights on 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.

Had a technical problem that prevented Zoom recording. Let me know if you have questions!

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

Zoom recording of Lecture 5 Passcode: 7R5!5fp=

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.

I woke up with a mild headache this morning, so I decided to work from home as a precaution. I will make up for this lecture in the future.

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

Zoom recording of Lecture 11 Passcode: kpjzvp6*

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

Zoom recording of Lecture 12 Passcode: &7nv&DwZ

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

Zoom recording of Lecture 13 Passcode: $#gJ79J5

### 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, *Enumerating odd-degree hyperelliptic curves and abelian surfaces over P ^{1}.* Mathematische Zeitschrift, 2023.

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.

__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.