Fall 13–Spring 14

**Professor:** Mark Haiman

**Office:** 855 Evans

**Spring Office hours:** WF 11:30-12:30 or by appointment

**Phone:** (510) 642-4318

**Prerequisites:** Math 250A or equivalent; knowledge of some
material from Math 250B also helpful. The fall semester (256A) is
required for the spring semester (256B).

**General:** This course is a two-semester
introduction to the foundations of algebraic geometry in the language
of *schemes*, along with techniques, examples and applications.

The theory of schemes was developed by Grothendieck and his
collaborators in the 1960's. The basics are set forth in Grothedieck's
treatise *Éléments de géométrie algébrique* (EGA), with more
advanced material presented in the various volumes of the *Séminaire
de Géométrie Algébrique* (SGA).

In many ways EGA and parts of SGA still provide the best exposition of the subject, if we update the treatment of sheaf cohomology (EGA doesn't use derived categories) and add something about Grassman schemes (omitted in EGA). A problem for English speaking students is that EGA and SGA are only available in the original French. My plan is to use EGA as the course text by providing synopses in English of the sections we cover, stating theorems, definitions and sometimes outlines of proofs.

In the lectures I will discuss proofs of the most important theorems, and attempt to illustrate and motivate the concepts with examples, something that EGA does not do.

**EGA:** A. Grothendieck, Éléments de géométrie algébrique,
I-IV, Publications Mathématiques de l'IHES, vols. 4, 8, 11, 17, 20,
24, 28, 32 (1960-1967).

It seems no longer possible to obtain EGA in print, but it is available electronically in the collections of IHES Volumes hosted by Springer or NUMDAM.

**SGA:** All volumes except SGA 2 were published in the Springer
Lecture Notes in Mathematics series. They are available
electronically through the UC libraries. The easiest way to access
them is from the
list
of references on Wikipedia. The entries there for the Springer
volumes contain links with labels like "doi:10.1007/BFb0058656," which
you click on the numeric string after "doi:". There are also entries
with math arXiv links for typeset versions of SGA 1 and SGA 2.

**Other references:** You may find some of the following English
language references helpful.

- Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press, 2002, 2006. The first 7 chapters give a general introduction to schemes much in the spirit of EGA. Available electronically through UC library.
- David Mumford The Red Book of Varieties and Schemes, Springer, 1999. An especially readable account, available electronically.
- David Eisenbud and Joe Harris, The Geometry of Schemes, Springer, 2000. Also highly readable, and available electronically.
- Robin Hartshorne, Algebraic Geometry Springer, 1997.

- David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, 2004.
- Michael Atiyah and I.G. Macdonald, Introduction to commutative algebra Addison-Wesley, 1969.

**Homework and grading:** Problems sets will be posted here from
time to time. I will assign more problems than I actuallly expect you
to do. Grading will be based on your doing a reasonable fraction of
them, although you are of course welcome to try to do them all if you
feel ambitious. Extra difficult problems are marked with an asterisk.

Lectures | Reading | Homework | Due |
---|---|---|---|

Jan 22-31 | EGA I, 2.5, 3.1-3.3, 5.1-5.2 | Problem set 6 | Feb 14 |

Feb 3-14 | EGA I, 3.4-3.6, 4.3-4.5, 5.3-5.5; EGA IV, 1.1-1.2 | ||

Feb 19-28 | EGA I, 1.6, 9.1-9.6; EGA II, 1.1-1.7 | Problem set 7 | Apr 25 |

Mar 3-April 9 | EGA II, 2.1-2.9 | ||

April 11-23 | EGA II, 3-5 | Problem set 8 | May 19 |

April 25-May 2 | Notes on sheaf cohomology |

Lectures | Reading | Homework | Due |
---|---|---|---|

Aug 30-Sept 9 | Examples in class | Problem set 1 (9/20: corrected typos in 1 and 2) | Oct 4 |

Sept 11-23 | Mumford "Red Book" I.2-I.5 | Problem set 2 | Oct 18 |

Sept 25-Oct 16 | EGA 0, 1-3.4; EGA I, 1.1-1.3 | Problem set 3 | Nov 1 |

Oct 18-Nov 1 | EGA 0, 3.5 & 4.1; EGA I, 1.7-1.8 & 2.1-2.4 | Problem set 4 | Nov 27 |

Nov 4-27 | EGA I, 6.1-6.4 & 6.6; EGA IV, 10.1-10.4 | Problem set 5 | Dec 19 |

Dec 2-Dec 6 | EGA I, 1.4 & 4.1-4.2 |

**EGA Synopses**

- Table of Contents
- Chapter 0, Sections 1-4
- Chapter I, Sections 1.1-1.3
- Chapter I, Sections 1.4-1.8
- Chapter I, Section 2
- Chapter I, Section 3
- Chapter I, Section 4
- Chapter I, Section 5
- Chapter I, Section 6
- Chapter I, Section 9
- Chapter II, Section 1
- Chapter II, Sections 2.1-2.4
- Chapter II, Sections 2.5-2.9
- Chapter II, Section 3
- Chapter II, Section 4
- Chapter II, Section 5
- Chapter IV, Sections 1.1-1.7
- Chapter IV, Sections 10.1-10.4

**Notes on Sheaf Cohomology**

**Syllabus:** I've divided the topics below into foundations and
applications. There are more topics on this syllabus, especially
toward the end, than we are likely to cover.

Foundations, Part I

- Overview of classical algebraic varieties
- Sheaves of functions; geometry (differential, analytic, algebraic) via structure sheaves
- Sheaves of rings and modules; functorial constructions
- Rings of fractions and localization
- Affine schemes
- Quasi-coherent sheaves on an affine scheme
- Schemes and morphisms; gluing
- The category of affine schemes; Spec as an adjoint functor
- Schemes over a ground ring or scheme
- Morphisms of finite type; algebraic schemes
- Jacobson schemes and Hilbert's Nullstellensatz
- Reduced and non-reduced schemes
- Schemes of finite type over a field
- Functors of points; geometric points
- How classical varieties are the same thing as reduced schemes locally of finite type over an algebraically closed field
- Coherent sheaves on Noetherian schemes

Foundations, Part II

- Products and base change
- Fibers of a morphism
- Surjective and injective morphisms
- Immersions
- Separated morphisms and separated schemes
- Functorial constructions for quasi-coherent sheaves
- Affine morphisms;
*Spec*of a quasi-coherent sheaf of*O*algebras_{X} - Vector bundles and locally free sheaves
- Projective schemes and Proj
- Projective morphisms and
*Proj* - Sheaves on
*Proj(A)* - Ample line bundles; functorial property of Proj
- Segre and Veronese morphisms
- Dimension
- Flat morphisms
- Regular rings and regular schemes
- Smooth and étale morphisms
- Differentials

Foundations, Part III

- Derived categories
- Sheaf cohomology
- Proper and projective morphisms
- Quasi-affine and quasi-projective morphisms
- Chow's lemma

Applications

- Projective space and projective varieties
- Curves, surfaces, hypersurfaces
- Tangent cones, tangent and normal bundles
- Algebraic groups, group actions, quotients
- Toric varieties
- Grassmann and flag varieties
- Nilpotent orbits, Springer varieties, Hessenberg varieties
- Hilbert schemes
- Chow varieties
- Divisors
- Blowing up
- Complex analytic techniques; Kodaira and Grauert-Riemenschneider vanishing theorems
- Characteristic
*p*techniques; Frobenius splitting - Intersection theory
- Riemann-Roch and Grothendieck-Riemann-Roch theorems
- Resolution of singularities
- Cohen-Macaulay and Gorenstein schemes, duality
- Rationality, unirationality and rational connectedness
- Azumaya algebras and the Brauer group