Our intent is to cover a canon completely and rigorously,
with enough examples and calculations to help develop intuition for the machinery.
This is often the content of a second course in algebraic geometry,
and in an ideal world,
people would learn this material over many years,
after having background courses in
commutative algebra, algebraic topology, differential geometry, complex analysis,
homological algebra, number theory, and French literature.
We do not live in an ideal world.
For this reason, the book is written as a first introduction, but a challenging one.
This book seeks to do a very few things, but to try to do them well...
The core of the material should be digestible over a single year.
After a year of blood, sweat, and tears,
readers should have a broad familiarity with the foundations of the subject,
and be ready to attend seminars, and learn more advanced material.
They should not just have a vague intuitive understanding of the ideas of the subject;
they should know interesting examples, know why they are interesting,
and be able to work through their details.
Readers in other fields of mathematics should know enough
to understand the algebro-geometric ideas that arise in their area of interest.
Basic Algebraic Geometry 1: Varieties in Projective Space.
Basic Algebraic Geometry 2: Schemes and Complex Manifolds.
third tome (about complex manifolds) won't be used in this course
There are at least 3 editions in Russian with the respective English translations:
Russian Original is spelled Основы алгебраической геометрии,
it has 3 editions:
1st in 1972 (Nauka, Moscow),
2nd in 1988 (Nauka, Moscow),
3rd in 2007 (МЦНМО, Moscow).
2nd English edition: 1994, Springer-Verlag.
Translated from Russian 1988 edition with notes by Miles Reid.
3rd English edition: 2013, Springer-Verlag,
ISBN 978-3-642-37955-0 (tome 1)
and 978-3-642-38009-9 (tome 2).
MR3100243.
This book is a general introduction to algebraic geometry. Its aim is a treatment of the subject as a whole, including the widest possible spectrum of topics...
The nature of the book requires the algebraic apparatus to be kept to a minimum. In addition to an undergraduate algebra course, we assume known basic material from field theory: finite and transcendental extensions (but not Galois theory), and from ring theory: ideals and quotient rings.
Cambridge University Press, 1993.
Algebraic geometry is a mixture of the ideas of two Mediterranean cultures.
It is the superposition
of the Arab science of the lightning calculation of the solutions of equations
over the Greek art of position and shape.
This tapestry was originally woven on European soil and is still being refined
under the influence of international fashion.
Algebraic geometry studies the delicate balance between the geometrically plausible and the algebraically possible.
Whenever one side of this mathematical teeter-totter outweighs the other,
one immediately loses interest and runs off in search of a more exciting amusement...
In this book we present from a modern point of view
the basic theory of algebraic varieties and their coherent cohomology.
The local part of the study includes dimension and smoothness.
I have tried to keep the commutative algebra down to minimum
while putting the geometry close to the algebra as part of the exposition.
The basic tools in algebraic geometry are sheaves and their cohomology.
This material is presented from the beginning.
I have included the basic discussion of curves to illustrate the theory...
...the main battle was to teach the reader to think globally in sheaf-theoretic language.
[09.08]
Modules over rings.
Bilinear maps.
Tensor products.
Explicit construction and as initial objects in the category of bilinear maps.
Kernels of bilinear maps.
Non-degenerate and perfect bilinear pairings.
Segre embedding.
Segre variety as variety of matrices (resp. bilinear forms or linear maps)
of rank one.
Inverse of Segre embedding via (co)images or (co)kernels.
Segre quadric and Riemann sphere.
Stereographic projection, its inverse,
indeterminacy loci and contracting sets.
Symmetric and skew-symmetric pairings.
Symmetric bilinear pairings as a quadratic form.
Vector spaces of forms
(homogeneous polynomials of degree d of n variables
are also called n-ary d-ic forms).
Veronese embedding.
Any smooth conic with a K-point
is projectively equivalent to the Veronose conic,
which can also be interpreted as
the locus of degenerate binary quadratic forms.
Interior and exterior of real Veronese conic,
its intersection with lines and numbers of real tangents.
[09.15] QA - base subring and base subfield, tensor product.
Cartesian product as an initial object.
Fiber-product. Pull-back.
Projective planes and projective duality.
Incidence variety = flag variety = universal line = universal point.
Synthetic approach: axioms of projective and affine plane, from projective to affine and back.
Projective spaces over skew-field, quaternionic projective space.
Pascal's theorem: six point lie on a conic iff three intersections of opposite sides are concurrent.
(see also the related theorems of
Pappus,
Brianchon,
Desargue;
later on we will also derive Pascal from Cayley-Bacharach).
Formulation: desarguean planes are in bijection with skew-fields.
Schedule change: from Sep 16 the classes are on Mondays and Wednesdays, 13:00-15:00.
[09.16]
[09.21]
[09.23] Categories and functors.
Examples: Sets, Cats, Rings, Top, Open(X).
Open as a functor from Top to Cats.
PreSheaves(X) = contravariant functors from Open(X).
First intuitive description of sheaves and of spaces-with-functions
(cf. [Kempf]).
Base of Zariski topology on affine spaces.
Induced Zariski topology on open and closed subsets.
Radical of an ideal, radical ideals.
Hilbert basis theorem: Noetherian rings and modules,
polynomials over Noetherian ring is Noetherian,
any ideal in finitely generated ring is finitely generated.
Announcement (Nullstellensatz) :
bijection between Zariski closed subsets of a complex affine space
and radical ideals in polynomial ring.
[09.28] {Yom Kipur - regular class}
Opposite category and contravariant functor Op.
Represented functors.
Natural transformations.
Category of functors.
[09.30] On ubiquity of functors
and natural transformation in geometry/topology.
Some history of math from 1820s to 1950s:
Poncelet, Abel, Jacobi, Salmon, Cayley,
Peano, C. Segre, Poincaré, Serre, Grothendieck.
Newton-Leibnitz-Cauchy-Stokes formula.
Chains, boundary maps, cycles, boundaries,
(Betti) homology. Betti numbers.
Example of torsion in homology: real projective plane.
Functions, differential 1-forms;
Grassmann algebra and higher differential forms;
de Rham differential,
closed and exact forms,
de Rham cohomology.
Integration as a pairing between cycles
and closed forms.
Pullback of functions and differential forms,
pushforward of chains.
Projection formula: the integral of a pullback equals
to the integral over a pushforward.
Homotopy category,
fundamental group, higher homotopy groups
and first cohomology as (co)representable functors.
Functor of abelianization from Groups to AbelianGroups,
relation between fundamental group and first homology.
Complexes and exact sequences,
homology is an obstruction to exactness.
Categorical POV, e.g.
(co)chains, functions and differential forms
as (co)functors from Top/Hot to Ab/Rings/Alg
and (co)boundaries as natural transformations,
etc.
[10.05] Polynomial endofunctors of Vect/k
(category of (finite-dimensional) vector spaces over a field).
Restrictions and more general pullbacks of homogeneous polynomials.
Intersection of rationally parametrized curves
with (hyper)surfaces in a projective space.
Formulation of Bezout theorem.
Parametrizanion of lines on surfaces:
dual projective plane.
base of a quadratic cone,
two families on a hyperboloid
(cf. Shukhov Tower
and hyperboloid structures).
Twenty seven lines on cubic surface ABC=XYZ, A+B+C = X+Y+Z = 0
and on Fermat cubic surface X^3 + Y^3 + Z^3 + W^3 = 0.
[10.07] Yoneda Lemma.
Fibrations, bundles, sheaves.
Modules over ring of functions.
Lozalization.
[10.14] {10.12 = N. Sra. Aparecida}
[10.19]
link to Rabinowitsch trick.
How many lines intersect four given lines in a space?
Sketch of two proofs: using Segre quadric (in P^3)
and using Plucker quadric (in P^5).
Grassmannian Gr(2,4) and its covering by 6 affine spaces A^4.
Exercise: can it be covered by 5 of these spaces? And by 4?
[10.21]
New abstract nonsense: adjoint functors;
adjointness of Forget and Free.
Vector bundles and Grassmannians.
Pullback (base change) of locally trivial fibration.
Exterior product of fibrations.
An example of a vector bundle
not isomorphic to its own
pullback with respect to an automorphism of a base.
(Unramied) coverings.
Reduction of a structure group in topology.
From tautological bundle over a projective line
to Moebius strip and Hopf fibration.
Group object of a category: two points of view.
Is functor of vector bundles representable?
[10.26] Vector bundles and sheaves.
Non-isomorphic vector bundles over circle with isomorphic pullbacks with respect to a double cover (topological example).
Operations in category of vector bundles:
direct sum, tensor product, determinant,
subbundles, quotient-bundles, short exact sequences.
Determinant of a direct sum is tensor product of determinants.
(same is true for short exact sequences, but I forgot to formulate).
In topology any subbundle is a direct summand,
in algebraic geometry this fails
(ex. - over a projective line,
consider the universal bundle as a subbundle
in trivial rank two bundle).
Global and local sections of the bundle, restriction of sections.
Three equivalent definitions of a sheaf.
Example 1 - sheaf of sections of a vector bundle.
Example 2 - sheav of localizations of an A-module M
over a spectrum of a commutative ring A.
Zariski topology on prime and maximal spectra Spec A and Spm A,
relation to Zariski topology of an algebraic set.
[10.28] Bonus: principal bundles.
[11.04]
Closure as a limit.
Geometric interpretation for addition of tensors: secant varieties.
Segre quadric surface and a homomorphism from PGL(2) x PGL(2) to PO(2,2).
Segre cubic threefold, twisted cubic curves,
and determinantal representation for their net of quadrics.
The projective spectrum of a homogeneous ring,
equipped with Zariski topology and the structure sheaf of graded algebras.
[11.09]
When two things are the same?
Isomorphisms from categorical point of view.
Algebraic, analytic, smooth, continuous and ''homotopic'' morphisms.
(Non-)existence of morphisms between affine and projective cubic curves - formulation and idea of the proof using periods.
Ringed spaces.
The structure sheaf of rings on the spectrum of a ring (explained for the case of a domain).
What is a scheme.
Intuition behind dimension and degree of a projective variety,
and an approach to define them using Hilbert function.
[11.11] Definition of regular sequence and complete intersection.
Nullstellensatz and correspondence between radical ideals and subvarieties.
Connectedness and idempotents.
Irreducibile and non-irreducible schemes.
Reduced and non-reduced schemes.
Krull theorem.
Local intersection index of two curves on a surface. Bezout theorem.
Twisted cubic curve.
[11.16] Valuations.
Ring, ideal and residue field of a discrete valuation;
uniformizers. Completion.
Discrete valuation rings (DVRs)
are regular local rings of dimension 1.
Divisors on complex projective line.
Principal divisors and divisor class group.
Degree of a divisor on CP^1.
Effective divisors.
Sheaf associated to divisor.
Example (for CP^1) of correspondence between classes
of (Cartier) divisors to isomorphism classes of line bundles
to invertible sheaves.
[11.18] Computation of global sections of line bundles on the projective line.
Example of Riemann-Roch formula on P^1.
Global sections of line bundles on projective spaces.
Canonical bundle of projective line.
[11.23] Cremona transformation of a plane. Fixed components
and based loci of rational maps. Graph of a rational map
and its closure. Movable and fixed parts of a linear system.
Birational isomorphism classes of irreducible varieties.
Rational and irrational varieties.
Rationality criterium for smooth conics.
Regularization of Cremona involution
to a regular involution of a blowup
of a plane in three points (del Pezzo surface of degree 6).
Formulation of Cremona theorem on generatedness
of the plane Cremona group by a single Cremona involution
and the group PGL(3,C) of biregular automorphisms.
Exercise: prove that any biregular automorphism of a projective space
is a projectivity, i.e. Aut_k P^n = PGL(n+1,k).
[11.25] The category of affine varieties (over (Spec) A)
and its equivalence to the category of A-algebras.
Category of schemes: objects and morphisms,
abstract and concrete points of view.
Push-forward of a sheaf (of algebras). Sheafification and étale space of a sheaf.
Exercise: compare push-forward of the structure sheaf of a closed subvariety Y
with the sheaf of quotient-rings with respect to the sheaf of ideals of I_Y.
[12.02] Coherence. Classes of morphisms. FLatness. Hilbert function and criterium of flatness in terms of Hilbert (exercise for exam).
[12.07] Relative Spec and Proj. Examples: covering ramified in a divisor,
projectivization of a vector bundle, blow-up of a sheaf of ideals.