# Summer Graduate School

Parent Program: |
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Location: |
Tsinghua Sanya International Mathematics Forum, China |

*[Image: The simplest interesting case of linkage (liaison) of curves in projective 3-space. We see two quadric surfaces, one of which is a cone, meeting in the union of a line (vertical in the illustration) and a twisted cubic (snaking up from the bottom left to the upper right, tangent to the line at the origin.]*

The theory of algebraic curves, arguably the oldest branch of algebraic geometry, has seen major developments in recent years, for example in the study of syzygies, and around questions about moduli spaces and Hilbert schemes of curves. The theory is rich in research activity and unsolved problems. There is an encyclopedic work by Arbarello, Cornalba, Griffiths and Harris, but there is no modern text that could be used as a textbook and that goes beyond the basics of the theory. We have embarked on a project to write a book at roughly the level of the wonderful book on complex algebraic surfaces by Arnaud Beauville. The intent can be seen from a list of some major topics it will treat:

- Linear series and Brill-Noether theory
- Personalities: curves in projective space with low genus and degree
- Overview of moduli and Jacobians
- Hilbert schemes
- Syzygies and linkage

The school will have two series of lectures, one by Harris and one by Eisenbud. Harris’ lectures will focus on the more geometric side of the theory, including Brill-Noether theory, families of curves and Jacobians; while Eisenbud’s lectures will focus on the more algebraic side of the theory, including properties of the homogeneous coordinate rings of curves (Cohen-Macaulay, Gorenstein, free resolutions, scrolls, ...) Both lecturers will rely on chapters from the forthcoming book, which should be finished in large part by the time of the school. In addition, some of Eisenbud’s lectures will treat the use of Macaulay2 to investigate the projective embeddings of curves.

**Suggested prerequisites**

Students should be comfortable with the following ideas, constructions, and results:

*Algebraic Geometry*

- Abstract varieties and subvarieties of projective space at the level of [5, Chapters 1–3 ].
- The relations between (rational) maps to projective space, divisors and divisor classes, line bundle and invertible sheaves at the level of [5, Chapter 8 and Appendix].
- Definition of cohomology, cohomology of line bundles on Pn[4].
- Serre’s theorems: vanishing of cohomology above the dimension; vanishing of higher cohomology of high twists [4] or [3, Ch 3].
- Riemann-Roch for curves. [3, Section IV.1].

*Commutative Algebra*

- Basic Commutative algebra at the level of [1].
- Some homological ideas: complexes, free resolutions, Hilbert’s Syzygy Theorem [2, Chapter 1].

*Bibliography*

[1] Atiyah, M. F. and McDonald, I. G. *Introduction to commutative algebra*. Addison-Wesley Publishing Co., Reading, Mass., 1969.

[2] Eisenbud, D., *Commutative algebra. With a view toward algebraic geometry*. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995

[3] Hartshorne, R., *Algebraic geometry*. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

[4] Serre, J.-P. *Faisceau Algebriques Coherents Ann. of Math*. (2) 61, (1955). 197–278. English Translation (Achinger) at https://achinger.impan.pl/fac/fac.pdf.

[5] Smith, K., Kahanpaa, L., Kekalainen, P., and Traves, W., *An invitation to algebraic geometry*. Universitext. Springer-Verlag, New York, 2000.

For **eligibility** and **how to apply**, see the **Summer Graduate Schools homepage**

**Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.**

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

Brill Noether theory

Castelnuovo Theory

curves

Jacobians

Scrolls

free resolutions

moduli

Hilbert Schemes

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**No Secondary AMS MSC