Graduate Algebraic Geometry Seminar Spring 2023
About
This is a graduate student seminar at UIC in algebraic geometry. We meet every Monday 4:00-5:00 pm at SEO 427.
Week 12: Apr 3, 2023
Sean Edwards : Geometry of Hurwitz scheme
Hurwitz schemes are schemes which parameterize branched covers of curves with fixed degree and number of branch points. In this talk, we will begin by defining them as sets and show that they have structures of complex manifolds and schemes. For a particular case, it can be shown that the Hurwitz scheme is connected. Using this, we can prove that the moduli space of curves of a fixed genus is irreducible. This talk is roughly half topology/complex analysis and half material at the level of Hartshorne ch. 3 and 4, so it should be very accessible to everyone.
Week 11: Mar 27, 2023
Junyan Zhao: Measuring degree of irrationality
Today we'll start talking about measures of degree of irrationality of (smooth) projective varieties as an extension of our course 571 this semester. I'll introduce some background information and important techniques to this problem.
Week 10: Mar 13, 2023
Ben Gould: Reider's Theorem
Disclaimer: this abstract has been AI generated, there could be some errors and inaccuracies.
Reider's Theorem is a fundamental result in algebraic geometry, which provides a powerful criterion for the existence of certain types of divisors on algebraic varieties. In this seminar talk, we will discuss the statement and proof of Reider's Theorem, as well as some of its applications in algebraic geometry, including the study of moduli spaces and the theory of algebraic surfaces. We will also explore some recent developments and generalizations of Reider's Theorem, including its relationship to other conjectures in algebraic geometry, such as the abundance conjecture. This seminar is aimed at graduate students and researchers interested in algebraic geometry, and assumes some familiarity with basic concepts such as divisors, sheaves, and cohomology.
Week 9: Mar 6, 2023
Yeqin Liu: Exceptional vector bundles on P^3 are determined by their Chern characters
Exceptional vector bundles are a special class of rigid vector bundles. In particular, the moduli space of exceptional bundles with a fixed Chern character is discrete. On P^3, it has been conjectured that such moduli space consists of a single point. In this talk, we prove this conjecture.
Week 8: Feb 27, 2023
Gregory Taylor: The Gonality Theorem for Curves
We discuss Ein and Lazarsfeld's proof of the Gonality Conjecture. We will present Voisin's interpretation of Koszul cohomology via the Hilbert scheme, the duality theorem, and the cohomological techniques required for the proof.
References:
Ein, L., & Lazarsfeld, R. (2014, July 16). The gonality conjecture on syzygies of algebraic curves of large degree. arXiv. http://arxiv.org/abs/1407.4445. Accessed 27 February 2023
Week 7: Feb 20, 2023
Junyan Zhao: Brill-Noether theory on curves via the vector bundle on K3 (II)
References:
Lazarsfeld, R. (1989). A Sampling of Vector Bundle Techniques in the Study of Linear Series. WORLD SCIENTIFIC. https://doi.org/10.1142/0851
Week 6: Feb 13, 2023
References:
Lazarsfeld, R. (1989). A Sampling of Vector Bundle Techniques in the Study of Linear Series. WORLD SCIENTIFIC. https://doi.org/10.1142/0851
Week 5: Feb 6, 2023
Henry Fontana: Normally Generated Line Bundles (II)
References:
Lazarsfeld, R. (1989). A Sampling of Vector Bundle Techniques in the Study of Linear Series. WORLD SCIENTIFIC. https://doi.org/10.1142/0851
Week 4: Jan 30, 2023
Henry Fontana: Normally Generated Line Bundles (I)
In this talk we will discuss the Clifford index of a curve. In particular we will prove a theorem that gives sufficient conditions for a line bundle to embed a curve into protective space as a projectively normal variety. The proof of this theorem relies on a the characterization of the failure of normal generation as the existence of a specific rank two vector bundle on the curve.
References:
Lazarsfeld, R. (1989). A Sampling of Vector Bundle Techniques in the Study of Linear Series. WORLD SCIENTIFIC. https://doi.org/10.1142/0851
Week 3: Jan 23, 2023
I will prove Green's theorem, which states a curve embedded by a line bundle of high degree has more "reasonable" syzygies as we increase the degree. This is the first talk in a series of talks where we explore Larzasfeld's note "A Sampling of Vector Bundle Techniques in the Study of Linear Series".
References:
Lazarsfeld, R. (1989). A Sampling of Vector Bundle Techniques in the Study of Linear Series. WORLD SCIENTIFIC. https://doi.org/10.1142/0851