# BAGELS 2024 Spring

BAGELS (Best Algebraic Geometry Eating and Learning Seminar) is a students-running algebraic geometry seminar. We meet weekly and share our knowledge and research on algebraic geometry and related areas. You can give a talk about what you are studying, reading, or researching. For past BAGELS talks, see BAGELS Spring 2023

During 2024 Spring semester, we meet on Wednesdays from 3:30-4:30 in JWB 308.

# Schedule

Date | Speaker | Title/Abstract |
---|---|---|

Jan. 24 | Organizational Meeting | |

Feb. 14 | Yu-Ting Huang | The Boundedness of Complements for Log Surfaces Abstract: Shokurov posed the idea of complements in his work, 3-fold Log Flips, 1992. Since then, the theory of complements has played an essential role in birational geometry. People are concerned about the existence as well as the boundedness of complements. In the past 20 years, lots of related works have been done. For example, Birkar proved that the complements are bounded for Fano type morphisms in his well-known paper, Anti-pluricanonical System on Fano Varieties, 2019. In this talk, I will define the complement and prove the existence and boundedness of log Fano type surfaces. |

Feb. 21 | Rahul Ajit | Bertini Properties over finite field Abstract: I will present Poonen’s Sieve methods and sketch a proof of Bertini for smoothness over finite fields. Time permitting, I’ll sketch a proof of Bertini property for irreducibility due to Charles-Poonen. Then I’ll give a survey of known result and mention open questions in this direction. Finally, I’ll briefly talk about my joint projects with Matthew Bertucci and Daniel Apsley. I won’t assume any prerequisites outside basic AG. |

Feb. 28 | Zach Mere | Arc spaces and the Nash problem Abstract: Nash viewed the theory of arc spaces as a tool for studying the singularities of a complex variety. In 1968, he predicted a bijection between families of arcs through singularities and so-called “essential” divisors, which appear as exceptional in every resolution of singularities. This problem was wide open until recently. The correspondence has since been shown to hold for surfaces, but to fail in general in higher dimensions. However, certain partial results have been proven true in all dimensions, suggesting that a slightly different formulation might work. In this talk, we’ll discuss this recent progress towards a solution to the Nash problem. |

Mar. 13 | Joseph Sullivan | A Toric Proof of Pick’s Theorem Abstract: Pick’s theorem is a cute result which tells you how to compute the area of a lattice polygon by counting. Though it has much simpler proofs, we use it as an excuse to take a tour through toric geometry. We introduce toric varieties, we sketch some proofs highlighting the correspondence between their combinatorial data and geometry, and finally we deduce Pick’s theorem (and more!) by studying the Hilbert polynomial of a divisor associated to our polygon. |

Mar. 20 | Yi-Heng Tsai | Notions of numerical Iitaka dimension Abstract: The Iitaka dimension of a Cartier divisor D on a smooth projective variety is an important invariant that measures the growth of the global of mD. However, it does not only depend on the numerical class of D. So, Nakayama defined the numerical Iitaka dimension k_sigma by perturbing mD by a fixed ample divisor. An interesting question would be whether the growth of the global of [mD]+A can be controlled by m^{k_sigma}. In this talk, following John Lesieutre’s paper, we’ll study a complete intersection of general divisors of bidegree (1,1), (1,1) and (2,2), which would be an illuminating example of the question. |

Mar. 27 | Daniel Apsley | Bertini Theorems, Field Extensions, and Zeta Functions Abstract: In this talk we continue the discussion of Bertini theorems over finite fields started by Rahul a couple of weeks ago. In particular we will cover the proof of Poonen’s Bertini theorem and review some algebraic geometry over finite fields, in order to talk about some of our progress towards understanding the behavior of this Poonen’s theorem under field extensions and restrictions. |

Apr. 10 | Will Legg | Ordered Blueprints, F_1 Geometry, and Berkovich Analytification Abstract: Recent progress in F_1 geometry has brought about striking connections to both tropical and analytic geometry over non-Archimedean fields. In this talk, we will introduce Lorscheid’s language of “ordered blueprints,” algebraic structures analogous to ordered semirings, to describe this relationship. As a consequence, we will gain a robust notion of F_1-schemes, as well as the recovery of Berkovich analytification from the points of certain moduli schemes valued in the tropical semiring. |

Apr. 17 | Qingyuan Xue | Complexity and Toric Varieties Abstract: Toric varieties are fundamental objects in algebraic geometry, known for their intricate interplay between combinatorial and geometric structures, which provide deep insights into diverse algebraic phenomena. In this talk, I will explore the notion of complexity derived from birational geometry and present a characterization of toric varieties using this concept. |