---
created: 2022-04-05T23:42
updated: 2024-02-21T17:26
aliases:
  - algebraic curve
  - algebraic curves
  - curve
  - curves
  - Prym
  - Prym variety
---
---
- Links:
	- [valuation](Unsorted/Valuations.md)
		- [Archimedean valuation](Unsorted/Valuations.md)
	- [fraction field](Unsorted/localization%20of%20rings.md)
	- [completion](Unsorted/adic%20completion.md)

---

# algebraic curve

![](2024-01-07.png)
![](2024-01-07-1.png)

# Notes

- Encode points on a smooth projective curve as valuations measuring orders of poles/zeros.
	- Bounded valuations: points on the variety
	- Unbounded: points "at infinity", like puncture points on a Riemann surface
- Compactifications are not unique. Example:
	- $\AA^2 \subseteq \PP^2$
	- $\AA^2 \subseteq (\PP^1)^{\times 2}$
	- But $\PP^2 \neq (\PP^1)^{\times 2}$!

The algebraic analogues of a compact [proper](proper) complex algebraic curve of genus $g$.

# Prym varieties

![](attachments/Pasted%20image%2020220502181119.png)

![](2024-02-21.png)
# Course Outline

See <https://math.berkeley.edu/~bernd/math255.html>.

- January 22: Foundations [Kirwan, Chapter 2]  
- January 24: Bezout's Theorem [Kirwan, Section 3.1]  
- January 29: Points of inflection and cubic curves [Kirwan, Section 3.2]  
- January 31: The degree-genus formula [Kirwan, Section 4.1]  
- February 5: Branched covers of the line [Kirwan, Sections 4.2-4.3]  
- February 7: [Invariant Theory](https://math.berkeley.edu/~seigal/invariant.html) of Plane Curves  
- February 12: The [Weierstrass p-function](http://www.math.brown.edu//~jhs/RPEC/RESNotes.pdf) [Kirwan, Section 5.1]  
- February 14: Riemann surfaces [Kirwan, Section 5.2]  
- February 19 [MB]: Holomorphic differentials [Kirwan, Section 6.1]  
- February 21 [MB]: Abel's Theorem [Kirwan, Section 6.2]  
- February 26: The Riemann-Roch Theorem [Kirwan, Section 6.3]  
- February 28: The Riemann-Roch Theorem [Kirwan, Section 6.3]  
- March 5: Local rings, DVRs, Multiplicities [Fulton, Sections 2.4, 2.5, 3.1, 3.2]  
- March 7: Linear Systems, Multiple Points, Noether's Theorem [Fulton, Sections 5.2, 5.4, 5.5]  
- March 12: Varieties, Morphisms, and Rational Maps [Fulton, Chapter 6]  
- March 14: Resolution of Singularities [Fulton, Chapter 7]  
- March 19 [LC]: Divisors, Riemann's Theorem, Differentials [Fulton 8.1-8.4]  
- March 21 [LC]: Canonical Divisors and Riemann-Roch [Fulton 8.5-8.6]

**Homework:** There are seven assignments. Click on the date to see solutions:  

- **due** **[January 29](https://math.berkeley.edu/~bernd/curveshw1.pdf):** Kirwan 2.2, 2.4, 2.5, 2.7, 2.8, 3.1, 3.6  
- **due [February 5](https://math.berkeley.edu/~bernd/curveshw2.pdf):** Kirwan 3.3, 3.8, 3.11, 3.13, 3.14, 3.16  
- **due [February 12](https://math.berkeley.edu/~bernd/curveshw3.pdf):** Kirwan 4.1, 4.2, 4.3, 4.4, 4.5  
- **due [February 19](https://math.berkeley.edu/~bernd/curveshw4.pdf):** Kirwan 5.4, 5.9, 5.10, 5.12, 5.14, 5.18  
- **due [February 26](https://math.berkeley.edu/~bernd/curveshw5.pdf):** Kirwan 6.1, 6.3, 6.5, 6.6, 6.7, 6.8  
- (Problem 6.3 has a typo: one occurrence of "meromorphic" should be "holomorphic")  
- **due March 5:** Kirwan 6.10, 6.11, 6.15 and Fulton 2.17, 8.2, 8.6  
- (Problem 6.15: must assume that the curve has genus one)  
- **due March 12:** Fulton 2.25, 2.28, 3.6, 3.14, 5.11, 5.19, 5.21, 5.21, 5.30