*
An updated version of the notes will be gradually posted here, starting roughly September 1, 2010. See this mathoverflow page for interesting related discussion.
*

*The table of contents (and sketchy introduction) for the notes as of June 15, 2010
(much of the course) is here.
Topics near the end are quite rough (see the bottom of this page for further discussion). I'll drop off copies in the mailboxes of many of the people in the class late in the week of June 15. A paper copy of the current version will be available outside my office then too. *

There are several types of courses that can go under the name of
"introduction to algebraic geometry": complex geometry; the theory of
varieties; a non-rigorous examples-based course; algebraic geometry
for number theorists (perhaps focusing on elliptic curves); and more.
There is a place for each of these courses. *This* course will deal
with schemes, and will attempt to be faster and more complete and
rigorous than most, but with enough examples and calculations to help
develop intuition for the machinery.
Such a course is normally a
"second course" in algebraic geometry, and in an ideal world, people
would learn this material over many years. We do not live in an ideal
world. To make things worse, I am experimenting with the material,
and trying to see if a non-traditional presentation will make it possible
to help people learn this material better, so this
year's course is only an approximation.
(See
here
for an earlier version.)

This course is for mathematicians intending to get near the boundary of current research, in algebraic geometry or a related part of mathematics. It is not intended for undergraduates or people in other fields; for that, people should take Maryam Mirzakhani's class, or else wait for a later incarnation of Math 216 (which will vary in style over the years).

In short, this not a course to take casually. But if you have the interest and time and energy, I will do my best to make this rewarding.

**Office hours:**
Because of the nature of this class, I'd like to be as open as possible about office hours, and not have them restricted to a few hours per week. So if you would like to chat, please let me know, and I'll be most likely happy to meet on a couple of days' notice. I am almost always available to meet immediately after class.

**References:**
I hope to periodically release notes,
perhaps once per week.
(The most recent version should be outside my office door.
I will also give each new version out in class as they become available.)
You should take notes yourself, and not count on these.
The notes from the class two years ago are available
here.
It may be useful having Hartshorne's
*Algebraic Geometry*, and possibly Mumford's *Red Book of Varieties and Schemes*
(the original edition is better, as Springer introduced errors
into the second edition by retyping it). Mumford (second edition) is availble online
(with a Stanford account) from Springer.
For background on commutative algebra, I'd suggest consulting
Eisenbud's *Commutative Algebra with a View toward Algebraic Geometry*
or Atiyah and MacDonald's
*Commutative Algebra*. For background on abstract nonsense,
Weibel's *Introduction to Homological Algebra* is good to have handy.
Freyd's *Abelian Categories*
is available online (free and legally) here.

**Homework:** Unlike most advanced graduate courses, there
will be homework. It is important --- this material is very dense,
and the only way to understand it is to grapple with it at close range.
There will be a problem set most weeks.
Your grade will depend on the problem sets.

**Fall quarter**

- Mon. Sept. 21: introduction. Why you shouldn't take this class. What I'm trying to do in this class. What algebraic geometry is about. That's too hard, so at least what this course is about: why many notions (geometric, arithmetic, algebraic, complex-analytic, ...) can be understood in terms of "geometric spaces", and constructions related to them. Example: Mordell's Conjecture (Faltings' Theorem). A little bit of category theroy: objects, morphisms, source, target, identity, isomorphism, automorphism, examples (sets, vector spaces, A-modules, abelian groups, rings, topological spaces, partially order sets, the open sets of a topological space, subcategory.
- Wed. Sept. 23: covariant functor, forgetfunctor, contravariant functor, opposite category, full and faithful functors. Universal properties: tensor product and its properties, initial object, final object, zero object, fibered product, monomorphism, epimorphism, coproduct.
- Problem set 1 due Friday Oct. 2.

- Fri. Sept. 25: small category, diagram indexed by an index category, limit (inverse limit, projective limit), colimit (direct limit, injective limit),adjoint functors, tensor product is adjoint to Hom for A-modules.
- Mon. Sept. 28: groupification and other adjoint constructions "projecting categories into smaller ones", additive category, homomorphism, additive functor, kernel, cokernel, monic, subobject, epi, quotient object, abelian category, image, complex, exact, (co)homology, long exact sequence, left-exact and right-exact (additive) functor, exact (additive) functor, fun facts about interactions of (co)limits, adjoints, and left/right-exactness.
- Wed. Sept. 30 ((pre)sheaves): motivation (differentiable or continuous or arbitrary functions on a real manifold), germ, stalk, presheaf, sections of a (pre)sheaf over an open set, sheaf, identity axiom, gluability axiom, restriction of a sheaf, skyscraper sheaf, pushforward (direct image) sheaf.
- Thurs. Oct. 1: Drop by my office 3:20-4:10 if you'd like to discuss
complexes, homology, or other abelian category related things.
- Problem set 2 due Friday Oct. 9.

- Fri. Oct. 2: constant presheaves, locally constant sheaves, sheaf of sections of a continuous map, direct image / pushforward sheaf, ringed spaces and O_X-modules, morphisms of (pre)sheaves, presheaves of abelian groups (etc.) form an abelian category, kernels of sheaves of abelian groups (and cokernel worries), sections are determined by germs, compatible germs, morphisms are determined by stalks.
- Mon. Oct. 5: sheafification (universal property; construction by compatible germs), subsheaves, quotient sheaves; sheaves of abelian groups and O_X-modules form abelian categories; left exactness of global section functor and pushforward; tensor products of O_X-modules; the inverse image sheaf (adjoint description and colimit construction), examples.
- Wed. Oct. 7: base of topology, sheaf on a base, sheaves on a base are same as sheaves (including morphisms). Toward schemes: set, topology, sheaf of functions, and more motivation from manifolds. Affine schemes: Spec A as a set,
and a preliminary example. Functions, and values at a point.
- Problem set 3 due Friday Oct. 16.

- Fri. Oct. 9: the affine line over various fields, Spec Z, affine n-space, the primes of C[x,y,z], Nullstellensatz (first version), maps of rings induce maps of spectra (as sets), examples (quotient rings, maps of affine complex algebraic varieties, localizations), nilpotents, functions aren't determined by their values at points, nilradical.
- Mon. Oct. 12: Zariski topology, vanishing set, radical, radical ideal, maps of rings induce continuous maps of spectra, distinguished open sets, preliminary Noetherian discusion.
- Problem set 4 due Friday Oct. 23.

- No class Oct. 14 or 16. Instead, read to the end of the chapter 3 notes given out on Mon. Oct. 12. Words you should be comfortable with (in this setting): irreducible, closed point, quasicompact, specialization, generization, generic point, Noetherian topological space, Hilbert basis theorem, A[[x]], irreducible component, Noetherian induction, connected, conected component, the function I from subsets of Spec A to ideals of A. Jack is happy to meet on Thursday (as well as his usual office hours); e-mail him if you'd like to set up a time.
- Mon. Oct. 19: the structure sheaf of Spec A (by defining the sections over D(f) as the localization of A at those functions nonvanishing in D(f), and showing this is a sheaf on the base); the O_{Spec A}-module M-tilde, isomorphism of ringed spaces, affine scheme, scheme, isomorphism of schemes, open subscheme, affine open subset/subscheme.
- Wed. Oct. 21: stalks of the structure sheaf; schemes are local ringed spaces; examples (plane minus origin, gluing schemes, affine line with doubled origin, the projective line, projective space).
- Problem set 5 due Friday Oct. 30.

- Fri. Oct. 23: topological properties of schemes: irreducible, irreducible component, closed point, specialization, gener(al)ization, generic point, connected, connected component, quasicompact as before; quasiseparated. Reducedness and integrality. The affine communication lemma.
- Mon. Oct. 26: (locally) Noetherian scheme, reducedness is affine-local, A-schemes (schemes over a ring A), (locally of) finite type A-schemes, normal, factorial.
- Wed. Oct. 28: associated points of schemes, and fuzzy mathematics; embedded point, rational function, domain of definition, regular at a point, , total fraction ring. Generic points are associated points; if X is reduced, then it has no embedded points; functions are determined by their germs at associated points; zero divisors are precisely those functions vanishing at an associated point of X Primary ideals, (minimal) primary decmposition (and its "uniqueness"), associated primes (of an ideal, of a ring).
- Problem set 6 due Friday November 6.

- Fri. Oct. 30: morphisms of schemes as maps of local-ringed spaces. Complex schemes, or more generally k-schemes (where k is a field), or more generaly A-schemes (A is a ring), or more generally S-schemes (S is a scheme).
- Mon. Nov. 2: picturing schemes (generic points, nonreduced behavior). Mining an example: maps to A^1_Z correspond to global functions. Generalization 1: representable (contravariant) functors, uniqueness of the representing object up to unique isomorphism, functor of points, A-valued points of a scheme. Generalization 2: group schemes (and more generally group objects in a category). Examples: A^n, G_m, mu_p.
- Wed. Nov. 4: group schemes via functors of points. Useful types of morphisms: quasicompact, quasiseparated, open immersion, affine.
- Problem set 7 due Friday November 13.

- Fri. Nov. 6: finite morphisms, closed immersions and closed subschemes, criterion for a sheaf of ideals to come from a closed subscheme.
- Mon. Nov. 9: locally principal closed subschemes (and effective Cartier divisors), closed immersions in projective space (hypersurface, degree, hyperplane, quadric, cubic, quartic, quintic, ..., line, conic curve), locally closed immersions/subschemes, scheme-theoretic image (determined affine-locally for affine morphisms or reduced source).
- Wed. Nov. 11: scheme-theoretic closure of a locally closed subscheme, induced (closed sub)scheme structure on a closd subset, reduced version (reduction) of a scheme, morphisms (locally) of finite type/presentation, quasifinite.
- Problem set 8 due Friday November 20.

- Fri. Nov. 13: projective schemes, projective coordinates, homogeneous ideal, finitely generated graded ring (idiosyncratic definition), irrelevant ideal, Proj construction, projective distinguished open set, (quasi-)projective A-scheme, (quasi-)projective k-variety.
- Mon. Nov. 16: affine and projective cone of a projective A-scheme; maps of graded rings and maps of projective schemes; linear space, line, plane, n-plane, hyperplane; analogues of results for affine schemes; the nth Veronese subalgebra of a graded ring, and the Veronese embedding; classifying plane conics; rational normal curves.
- Wed. Nov. 18: examples: Veronese embedding, rulings on the quadric surface, weighted projective space. Fibered products: the building blocks (affines; open immersions); fibered products exist. In notes: Reinterpretation of the existence argument in terms of representable functors.
- Fri. Nov. 20: computing fibered products in practice; pulling back families; fibers of morphisms; every reasonable property is preserved by base change; how to fix properties not preserved by base change (geometric points, geometric fibers, geometrically connected, geometrically irreducible, geometrically reduced); products of projective A-schemes are projective A-schemes (the Segre embedding). Here is a bonus handout on properties of geometric fibers, beyond the scope of the course (small correction as of Oct. 18 2010).
- Problem set 9 due Friday December 4.

- Mon. Nov. 30: images of morphisms: elimination of parameters, Chevalley's theorem, fundamental theorem of elimination theory.
- Wed. Dec. 2: separated morphisms (examples: open, closed immersions, monomorphisms, affine to affine, projective space). Quasiseparated morphisms done right (examples: maps from Noetherian schemes; separated moprhisms). These notions behave well (preserved by base change; local on target; closed under composition; closed under products). Varieties.
- Fri. Dec. 4: More on separatedness. Proper morphisms.

- Mon. Jan. 4: rational maps, domain of definition, graph of a rational map, dominant rational map, birational maps and morphisms, rational varieties, ratinality of some plane conics, Cremona transformation, example of blowing up.
- Wed. Jan. 6: integral homorphisms and extensions of rings, and morphisms of schemes; finite implies integral, transcendence theory, going-up theorem, Nakayama's lemma (many versions), normalization (exists) (in a finite field extension), examples, finiteness of integral closure. Dimension (of a topological space or ring), curve, surface.
- Fri. Jan. 8: codimension, hypersurface, dimension=transcendence degree, Noether normalization, Krull's principal ideal theorem, pathologies of codimension, Algebraic Hartogs' lemma, UFD iff all codim 1 primes are principal.
- Problem set 10 due Friday January 22.

- Wed. Jan. 13 nonsingularity ("absolute smoothness" of Noetherian schemes), taught by Brian Conrad: Zariski tangent space and why it deserves the name.
- Wed. Jan. 20: the local dimension is at most the dimension of the cotangent space, the Jacobian criterion, Euler test, two facts stated but not proved (regular local rings are UFDs, and remain regular upon localization; won't be used). Seven faces of Discrete Valuation Rings.
- Fri. Jan. 22: valuative criteria (stated but not proved, won't be used). New topic (toward quasicoherent sheaves): vector bundles and locally free sheaves; transition matrices. Invertible sheaves. Properties of locally free sheaves.
- Problem set 11 due Friday January 29. Thanks to Ilya for a number of corrections to the notes!

- Wed. Jan. 27: topics I forgot (geometry of normal Noetherian schemes; orders of poles and zeros, Dedekind domains, Serre's criterion for normality). The distinguished affine base, and sheaves on them. (If you want to scare your friends, you can use the words: Grothendieck topology, site, topos.) Quasicoherent sheaves (2 equivalent definitions).
- Fri. Jan. 29: affine locality of quasicoherence. Pushforwards of quasicoherent sheaves by nice (quasicompact quasiseparated) morphisms are quasicoherent. Quasicoherent sheaves form an abelian category, and you can work affine-by-affine for all module-like constructions (tensor, Sym, wedge).
- Problem set 12 due Friday February 5.

- Wed. Feb. 3: torsion-free sheaves, quasicoherent sheaves of ideals = closed subschemes. Finiteness conditions: finite type and coherent sheaves. Geometric Nakayama, support, rank of a quasicoherent sheaf.
- Fri. Feb. 5.
- Problem set 13 due Friday February 19.

- No class week of Feb. 8-12.
- Wed. Feb. 17: invertible sheaves (line bundles). Some line bundles on projective space (O(n)). Invertible sheaves and Weil divisors: Weil divisors, irreducible, effective, support; regular in codimension 1, div = divisor of zeros and poles; O(D). The Picard group of projective space, hypersurface complements, quadric surfaces. (Nagata's lemma.) Effective Cartier divisors (= invertible ideal sheaves).
- Fri. Feb. 19: quasicoherent sheaves on projectie A-schemes and projective modules. Getting a quasicoherent sheaf from a graded module (the "tilde" functor). O(n), and shifting of the index of a graded module. Globally generation, and Serre's theorem. Getting from quasicoherent sheaves to graded modules: the Gamma_* functor, which is adjoint to tilde. Every quasicoherent sheaf on Proj A arises form the tilde construction. Saturated modules.
- Mon. Feb. 22 (pushforwards and pullbacks of quasicoherent sheaves): analogy with modules and ring maps; pushforwards of quasicoherent sheaves are quasicoherent (for quasicompact quasiseparated morphisms); pushforwards of coherent sheaves are coherent under finite maps; pullbacks (3 definitions) and their properties.
- Wed. Feb. 24: Maps to projective space via invertible sheaves. Relative ("sheafy") Spec.
- Fri. Feb. 26: Revisiting the construction of pullback and relative Spec. Relative Proj, and projective morphisms. (Sample consequence: an integral curve over a field has a birational model that is nonsingular and projective.)
- Problem set 14 due Friday March 5.

- Wed. Mar. 3 (introduction to cohomology): statement of desired properties. Consequences: cohomology of coherent sheaves on projective A-schemes are coherent A-modules (e.g. vector bundles on complex projective varieties have a finite-dimensional space of global sections); Serre vanishing; the only functions on projective intral schemes over algebraically closed fields are constants; pushforwards of coherent sheaves are coherent; finite = projective + affine; etc. Definition of Cech cohomology of a quasicoherent sheaf on a quasicompact seaprated scheme
*with respect to a finite affine cover*. - Fri. Mar. 5 (basics of cohomology): Cech cohomology is independent of affine cover. Proof of seven fundamental properties, including cohomology of the line bundles O(m) on projective space.
- Problem set 15 due Friday March 12.

- Mon. Mar. 8 (applications of cohomology to projective k-schemes): Euler characteristic (additive in exact sequences), Riemann-Roch for line bundles (or coherent sheaves) on a curve, degree of a line bundle or coherent sheaf on a curve, rank Hilbert function, Hilbert polynomial (of a coherent sheaf F has degree equal to the dimension of the support), Bezout's theorem (intersecting a hypersurface with any subvariety), arithmetic genus.
- Wed. Mar. 10: intersection theory on a nonsingular surface. Higher direct image sheaves = higher pushforward sheaves.
- Fri. Mar. 12: spectral sequences.

- Wed. Mar. 31: Tor and its properties; derived functors; fun with derived functors and spectral sequences.
- Wed. Apr. 7: more derived functors; Z-modules, A-modules, and O-modules have enough injectives; Cech cohomology = derived functor cohomology.
- Fri Apr. 9: concluding "Cech=derived functor" (Olsson's explanation of why quasicoherent sheaves on affine schemes have no derived functor cohomology); initial discussion of curves.
- Problem set 16 due Friday April 16.

- Mon. Apr. 12: the "Clear-denominators" extension theorem (from smooth curves to projective schemes); various "categories of curves" are all equivalent.
- Wed. Apr. 14: degree of a morphism between nonsingular curves.
To set up a discussion of curves: discussion of differentials and Serre duality (e.g. the degree of the canonical bundle is 2
*g*-2). - Mon. Apr. 19: the Riemann-Hurwitz formula, and a criterion for a morphism to be a closed immersion.
- Wed. Apr. 21: a series of crucial observations in preparation for the study of curves. (Conclusion: if L has degree at least 2g, it gives a morphism to projective space; if it has degree greater than 2g, it gives a closed immersion.) Curves of genus 0. The trichotomy of curves (genus 0, 1, and greater than 1), and how it generalizes.
- Mon. Apr. 26: curves of genus 2; hyperelliptic curves; curves of genus 3, 4, and 5.
- Problem set 17 due Wednesday May 12.

- Wed. Apr. 28: elliptic curves: classifying them, and the Weierstrass normal form.
- Wed. May 5: elliptic curves are group varieties. Fun applications to classical geometry.
- Fri. May 7: more classical geometry; specializing results (e.g. elliptic curves to Pascal to Pappus).
- Mon. May 10: (counter)examples using elliptic curves (a factorial scheme with every affine open not UFD; a crazy Picard group; an affine open of an affine that is not distinguished; a variety with non-finitely-generated space of global sections; a proper nonprojective surface).
- Wed. May 12: differentials.
- Fri. May 14: differentials and the Riemann-Hurwitz formula.differentials conti.
- Wed. May 19: Bertini's theorem and proof.
- Fri. May 21: the conormal exact sequence for nonsingular varieties.
- Wed. May 26: flatness: motivation, definition and easy facts, dimension in flat families, faithful flatness, flatness through Tor.
- Fri. May 28: flatness: ideal-theoretic criteria; flatness implies constant euler characteristic; semicontinuity, Grauert, and cohomology and base change. Mentioned: the local criterion, and consequences (slicing criterion, cohomology and base change).
- Mon. May 31: smooth, etale, unramified. The only hard fact: the conormal exact sequence is exact on the left if X/Y is smooth.
- Wed. June 2: proof of Serre duality for curves (and, with black boxes of miracle flatness and cup product, in higher dimension). The dualizing sheaf for curves is an invertible sheaf (a line bundle). Sketch of why it is the sheaf of differentials.
- Thurs. June 3, at noon (in the Faculty Area Research Seminar Series): plausibility argument that the moduli space of curves of genus g greater than 2 has dimension 3g-3.

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