This course provides an introduction to algebraic geometry. Algebraic
geometry is a central subject in modern mathematics, and an active area
of research. It has connections with number theory, differential
geometry, symplectic geometry, mathematical physics, string theory,
representation theory, combinatorics and others.
Math 203A covered affine and projective varieties. Math 203B covered
schemes and sheaf cohomology. Math 203C will continue from where Math
203B left off. Ample line bundles, flatness, curves, Chow groups, Chern classes.
Tu 2-3:30pm in AP&M 6-101. I am
available for
questions after
lecture or by appointment. Also, feel free to drop in if you see me in my
office.
Textbook: I will roughly follow
Andreas Gathamnn's notes available
online.
I recommend that you also consult Hartshorne's Algebraic
Geometry or Vakil's The Rising Sea.
There will be no exams for this class. The grade
will be
based entirely on homeworks and regular attendance of lectures. The
problem sets are mandatory and are a very important part of the course.
The problem sets are submitted via Gradescope.
Important dates:
First class: Tuesday, April 4
University Holiday: Monday, May 29
Last class: Thursday, June 8
Lecture Summaries
Lecture 1: Review of Math 203b. Cohomology of projective space and of complete intersections, glimpses of Serre duality. Curves. Riemann-Roch, Kodaira vanishing and statement of Serre duality. Equality of genera from Serre duality.
Lecture 2: Genus of nodal curves. Pullback of divisors. Degree of morphisms between curves. Topological and algebraic form of Riemann-Hurwitz. Topological genus.
Lecture 3: Finishing the proof of Riemann-Hurwitz. Cohomology of line bundles over curves. Very ample line bundles and projective embeddings. Ampleness over curves. Classification of curves of genus 0 and 1.
Lecture 4: Criterion for line bundles over curves to be basepoint free, very ample, ample. The canonical bundle is basepoint free. Non-hyperelliptic curves and very ampleness of the canonical bundle. Curves of genus 2 and 3.
Lecture 5: The canonical morphism for hyperlliptic curves. Classification of curves of genus 4 and 5. Genus 1 curves and the Abel map. The group law on the elliptic curve.
Lecture 6: Serre duality. Duality for higher rank bundles over the projective space. The existence of a dualizing sheaf. The case of curves and some ideas behind the argument.
Lecture 7: Serre duality for curves. Residues. Residue theorem. Proof of Serre duality.
Lecture 8: Flatness. Algebraic discussion of flatness. Examples and geometric visualization.
Lecture 9: Associated points of schemes. Flatness over regular curves and associated points. Existence of flat limits and an example. Invariants preserved by flatness: Euler characteristic, dimension, degree, genus.
Lecture 10: Moduli functors and moduli schemes. The Hilbert scheme. Example. The moduli of curves is not representable by a scheme.
Applications of flatness: infinitesimal deformations and tangent space to the Hilbert scheme.
Lecture 11: Higher pushforwards. Cohomology and flat base change. Semicontinuity theorem. Grauert's theorem. Cohomology and base change.
Lecture 12: Introduction to Chow groups. Rational equivalence. Motivation, definition and first examples.
Lecture 13: Properties of Chow groups. Excision, Mayer-Vietoris, affine bundles. Discussion of proper pushforwards and flat pullbacks.
Lecture 14: Proper pushforward and rational equivalence. Examples: projective space, affine stratifications, blowups of projective plane.
Lecture 15: A discussion of intersection products. Intersections with Cartier divisors. Self intersection of divisors over surfaces. Genus formula.
Lecture 16: Grassmannians and Schubert cycles. Kleiman-Bertini. Two lines intersecting four general lines.
Lecture 17: Projection formula. Segre and Chern classes defined. First properties of Segre and Chern classes.
Lecture 18: More on Segre and Chern classes. Whitney formula. Some examples.
Lecture 19: The splitting principle. Chern roots. Some examples.
Lecture 20: Cycles representing the top Chern class. Applications: 27 lines on the cubic surface, 2875 lines on quintic 3-fold. Further discussion of the geometry of cubic surfaces.