Flag varieties: projective spaces, Grassmannians, varieties of complete flags.
Schubert cells and cycles. Cohomology ring of varieties of complete flags: Borel presentation.
Schubert calculus: Pieri-Chevalley formula, divided difference operators.
Toric varieties: complex tori, affine and projective spaces, blow-ups of projective spaces.
Laurent polynomials and Newton polytopes. Kushnirenko and Bernstein-Khovanskii theorems
on the number of common zeroes of Laurent polynomials with given Newton polytopes.
Cohomology rings of smooth projective toric varieties: Pukhlikov-Khovanskii presentation.
Comparison of flag varieties and toric varieties. Gelfand-Zetlin polytopes as Newton polytopes
for flag varieties. Generalization of toric varieties: wonderful compactifications of De Concini-Procesi,
regular compactifications of reductive groups. Brion-Kazarnovskii theorem (extension of Kushnirenko
theorem). Newton polytope of regular compactification.
Intersection theory Notes to my course on intersection theory that I taught in Jacobs University Bremen in 2008.
These notes can serve as an introduction to the intersection theory on algebraic varieties.
We discuss in detail how to compute the intersection index of hypersurfaces. Main examples include
Schubert calculus on Grassmannians and Bezout theorem
for projective spaces and for toric varieties.
Recommended reading:
I. N. Bernstein, I. M. Gelfand, S. I. Gelfand,
Schubert cells, and the cohomology of the spaces $G/P$,
Russian Math. Surveys {\bf 28} (1973), no. 3, 1--26 This classical paper contains a description of polynomials (later called
Schubert polynomials) corresponding to Schubert cycles
in the Borel presentation.
There is also a helpful introduction to this paper.
M. Brion,
Lectures on the geometry of flag
varieties,
Topics in cohomological studies of algebraic varieties, 33--85, Trends Math., Birkhauser, Basel, 2005
Contains many useful facts on flag varieties.
Ph. Griffiths, J. Harris, Principles of
algebraic geometry. Reprint of the 1978 original.
Wiley Classics Library. John Wiley and Sons, Inc., New York, 1994
An exellent very well written book on complex algebraic geometry.
M.Demazure, Desingularization des Varietes de Schubert generalisees, Ann. Sc. Ec. Norm. Super. 7 (1974)
53-88 This is another classical paper describing Schubert polynomials. However, the approach is very different
from that of Bernstein-Gelfand-Gelfand.
S. Kleiman, D. Laksov,
Schubert Calculus,
Amer. Math. Monthly, 79 (1972), 1061-1082 A nice elementary introduction to intersection theory on Grassmannians.
A. Kirillov, S. Fomin,
The Yang-Baxter equation, symmetric functions, and Schubert polynomials,
Discrete Mathematics, 153 (1996), 123-143 A combinatorial description of monomials in a Schubert polynomial in terms of reduced diagrams
(pipe-dreams).
L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci.
SMF/AMS Texts and Monographs 6, 2001; Chapter 3 on Grassmannians and flag varieties
A very interesting and well-written book. The 3d chapter
deals with geometry of Grassmannians and of complete flag varieties.
I. R. Shafarevich, Basic algebraic geometry. 1. Varieties in projective space,
Second edition. Springer-Verlag, Berlin, 1994; Chapter on intersection indices
Introduction to the intersection theory from an algebraic viewpoint
Week by week details
Lecture 1
Schubert problem on four lines: how many lines intersect four given ones
in a 3-space? Enumerative geometry and Schubert calculus. Laurent polynomials and their Newton polytopes. Kushnirenko's theorem on the number of common zeroes
of Laurent polynomials in a complex torus (without proof yet). Problem on five conics:
how many conics are tangent to five given ones? Short overview of main examples of spherical varieties:
Grassmannians, varieties of partial flags, toric varieties associated with Newton polytopes, wonderful
compactification of the space of non-degenerate conics.
Lecture 2
Intersection theory: how to compute the intersection index ofn hypersurfaces on an
n-dimensional variety? Divisors, linear equivalence. The exceptional divisor on a blow-up
of the projective plane and its self-intersection index.
Hyperplane sections, degree of a projective variety. Picard group, effective and very ample divisors.
Solutions to the Schubert problem on 4 lines and to the problem on the number of conics
tangent to 3 given lines and passing through 2 given points.
Lecture 3
Variety of complete flags in C^n: action of GL_n, Schubert cells and cycles, Weyl group,
Bruhat decomposition. Example: variety of complete flags in C^3.
Generalized varieties of complete flags. Picard group and cohomology ring of the variety of complete
flags (Borel presentation). Projective bundle formula.
Lecture 4
Chern classes of vector bundles, Whitney sum formula. Line bundles and their Chern classes.
Proof of Borel presentation. Flag bundle theorem (a generalization of Borel theorem and of
projective bundle formula) and its proof. very ample divisors on the flag variety and a relation
with representation theory of the group GL_n.
Lecture 5
Schubert polynomials and divided difference operators (Demazure operators). Demazure's approach:
geometric meaning of divided difference operators. Bernstein-Gelfand-Gelfand approach: algebraic
and geometric Chevalley formulas.
Reduced diagrams (pipe-dreams) and Kirillov-Fomin theorem.
Lecture 6
Cohomology of varieties of partial flags. Borel theorem for generalized complete flag varieties.
Weight polytopes, permutohedron. Geometric realization of Schubert cycles, Demazure modules.
Self-duality of the basis of Schubert cycles and non-negativity of structure coefficients.
Chevalley formula in terms of weight polytopes.
Lecture 7
Toric varieties. Description through polytopes and fans. Correspondence between orbits and faces,
affine charts. Resolution of singularities of the weighted projective plane and relation with
continued fractions.
Lecture 8
Cellular decomposition for smooth toric varieties and Morse theory on Newton polytopes.
Chern classes of smooth toric varieties. Toric Fano varieties. Picard group and relation between
divisors and polytopes. Proof of Kouschnirenko theorem.
Lecture 9
Mixed volumes and Bernstein-Khovanskii theorem. Volume polynomial and
Puklikov-Khovanskii presentation for the cohomology ring of a smooth toric variety.
Comparison with Borel presentation for the cohomology ring of the variety of complete flags.
Intersection theory on homogeneous spaces: Kleiman transversality theorem, the ring of conditions
of a spherical homogeneous space.
Lecture 10
De Concini-Procesi theorem: the ring of conditions
of a spherical homogeneous space as a direct limit of the cohomology rings of smooth equivariant
compactifications. The ring of conditions of a complex torus via piecewise-polynomial continuous functions
on fans. Wonderful and regular compactifications of homogeneous spaces. Complete conics.
Lecture 11
Solution to the Chasles problem about the number of conics tangent to five given ones. Demazure
construction of a wonderful compactification. Regular compactifications of reductive groups
associated with representations. Relation with weight polytopes and with projective toric varieties.
Bialynicki-Birula cell decomposition.
Lecture 12
Chern classes of regular compactification. Picard group
and Brion-Kazarnovskii theorem. De Concini-Procesi algorithm and proof of Brion-Kazarnovskii theorem.
Gelfand-Zetlin polytopes and Newton polytopes of spherical varieties. Open problems.