Remedios Varo Vampiros vegatarianos (1962)
Remedios Varo Vampiros vegatarianos (1962)
Time: Tuesdays 11:30–12:50 & 13:30–14:50, Thursdays 11:30–12:50
Venue: KSE Dragon Capital building, Shpaka 3, room 4.07 (see KSE schedule for room changes)
Office hours: Thursdays, 10-11:30, Math department coworking space (floor 5.5), or by appointment
Algebraic geometry is the study of algebraically defined geometric objects, such as algebraic varieties (roughly, zero sets of systems of polynomial equations in many variables, over an algebraically closed field) or schemes (spaces which locally “look like” the prime spectrum of a commutative ring. It has a close relationship with commutative algebra, differential geometry, complex analysis, number theory, and representation theory.
The goal of the course is to introduce the audience to the basic objects of study in algebraic geometry: algebraic varieties and schemes, and (sheaves of) modules over them. Emphasis will be put on concrete objects such as projective varieties. In order to keep the course focused, we will work toward a single main goal, which will be the proof of the Weil conjecture (Riemann hypothesis) for curves over a finite field.
Syllabus: [PDF]
External links: Moodle KSE Hub Slack Schedule
Commutative algebra cheat sheet: [PDF]
Problem sets: 1 (due Jan 25)
| Jan 13 | I.1 Introduction. Algebraic sets in $\mathbb{A}^n$ and $\mathbb{P}^n$. |
| Jan 15 | I.2 Irreducible components, morphisms, spaces with functions. |
| Jan 20 | I.3 Products of varieties. Elimination theory. |
| Jan 22 | I.4 Tangent spaces and nonsingular varieties. Smooth curves. |
| Jan 27 | II.1 Intro to schemes. Sheaves |
| Jan 29 | II.2 Schemes and varieties |
| Feb 3 | II.3 Morphisms of varieties |
| Feb 5 | II.4 Separated and proper varieties |
| Feb 10 | II.5 Coherent sheaves |
| Feb 12 | II.5 Examples of coherent sheaves |
| Feb 17 | III.1 Cohomology via flabby resolutions |
| Feb 19 | III.2 Cohomology of affine schemes |
| Feb 24 | III.3 Cohomology of projective spaces |
| Feb 26 | IV.1 Intro to curves. Valuation criteria |
| Mar 3 | no class |
| Mar 5 | no class |
| Mar 10 | IV.2 Riemann–Roch and corollaries |
| Mar 12 | IV.3 Proof of Riemann–Roch |
| Mar 17 | IV.4 Elliptic curves |
| Mar 19 | V.1 Intro to surfaces. Intersection theory |
| Mar 23 | no class |
| Mar 25 | no class |
| Mar 31 | V.2 Riemann–Roch for surfaces and the Hodge index theorem |
| Apr 2 | V.3 Application to point counting on curves |
| Apr 7 | Bonus lecture |
| Apr 9 | no class |
| Apr 13 | no class |
| Apr 15 | exam |
Contact: pachinger@kse.org.ua