Algebraic Geometry Seminar

Department of Mathematics
Johns Hopkins University


Spring 2019

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 4:00)
Place: Shaffer 303

Date Speaker Title
January 29 Jingjun Han
Johns Hopkins
Complements and minimal log discrepancies (1)
Shokurov introduced the theory of complements while he investigated log flips of threefold. The theory is further developed by Prokhorov-Shokurov and Birkar. As one of key steps in the proof of BAB conjecture, Birkar showed a conjecture of Shokurov, i.e., the existence of n-complements for Fano varieties with pairs of hyperstandard coefficients. In this talk, I will show a more general version of his result and show the complements have some additional properties: divisibility, rationality, approximation, anisotropic. This is an ongoing work with Jihao Liu and Shokurov.
February 5 Victor Przyjalkowski
Steklov Mathematical Institute
Weighted complete intersections
We observe a classification and the main properties of one of the main class of examples of higher dimensional Fano varieties --- smooth complete intersections in weighted projective spaces. We discuss their main properties and boundness results. We also discuss extremal examples from Hodge theory point of view and their relations with derived categories structures and their semiorthogonal decompositions. If time permits, we discuss mirror symmetry for the complete intersections and invariants of their Landau--Ginzburg models related to ones of the complete intersections.
February 12 Jingjun Han
Johns Hopkins
Complements and minimal log discrepancies (2)
In this talk, I will continue to show the existence of complements. If time permits, I will give a sketch on how to prove ACC for minimal log discrepancies for a class of singularities using the theory of complements.
February 19 Ziwen Zhu
Utah
Higher Codimensional Alpha Invariants and Characterization of Projective Spaces
Recent work of Kento Fujita, Yuji Odaka and Chen Jiang shows that among K-semistable Fano manifolds, the projective space can be characterized in terms of either the alpha invariant or the volume. In this talk, I will generalize the definition of alpha invariant to arbitrary codimension, and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. This result also demonstrates the relation between alpha invariants and volumes in the characterization problem of projective spaces among K-semistable Fano manifolds.
February 26 Botong Wang
Wisconsin
Kazhdan-Lusztig theory for matroids
Matroids are basic combinatorial objects arising from graphs and vector configurations. Given a vector configuration, I will introduce a “matroid Schubert variety” which shares various similarities with classical Schubert varieties. I will discuss how the Hodge theory of such matroid Schubert varieties can be used to prove a purely combinatorial conjecture, the “top-heavy” conjecture of Dowling-Wilson. I will also report an on-going project joint with Tom Braden, June Huh, Jacob Matherne, Nick Proudfoot on the cohomology theory of non-realizable matroids.
March 5 Yajnaseni Dutta
Northwestern
Fujita type conjectures for pushforwards of pluricanonical sheaves
Extending the property that a line bundle on a smooth projective curve is globally generated if its degree is bigger than 2g, Takao Fujita, in 1985, conjectured that there is an effective bound on the twists by an ample line bundle to obtain global generation for canonical bundles. Even though the conjecture remains unsolved as of today, based on Demailly's singular divisor techniques, partial progress was made by Angehrn-Siu, Ein-Lazarsfeld, Heier, Helmke, Kawamata, Reider, Ye-Zhu et al. In this talk I will focus on similar global generation conjecture due to Popa and Schnell for pushforwards of canonical and pluricanonical bundles under certain morphisms f: Y --> X. The canonical bundle case first appeared in Kawamata's work in 2002 and the proof used Hodge theoretic techniques combined with the Demaiily's singularity techiniques. In this talk I will present a generic global generation result for log canonical pairs building on Kawamata's theorem. I will also discuss weak positivity properties of these pushforwards and its implications toward subadditivity of Kodaira dimensions. Some parts of this work was done jointly with Takumi Murayama.
March 26 Takumi Murayama
Michigan
Kalina Mincheva
Yale
Seshadri constants for vector bundles and applications
We introduce Seshadri constants for line bundles in a relative setting. They are a generalization of notions for line bundles and vector bundles respectively due to Demailly and to Beltrametti--Schneider--Sommese and Hacon, which were introduced to study Fujita's conjecture and variants thereof. We give three applications: (1) A characterization of projective space in terms of the Seshadri constant of the tangent bundle; (2) An identification of new nef classes on self-products of curves; and (3) A generic jet separation statement for direct images of pluricanonical bundles. This is joint work with Mihai Fulger.
The Picard group of a tropical toric scheme
Through the process of tropicalization one obtains from an algebraic variety $X$ a combinatorial object called the tropicalization of $X$, $trop(X)$, that retains a lot of information about the original variety. Following the work of J. Giansiracusa and N. Giansiracusa, one can endow $trop(X)$ with more structure, to obtain a tropical scheme. Loosely speaking, we consider more equations than the ones needed to determine the tropical variety. We are interested what information about the original variety $X$ is preserved by the tropical scheme $X_\mathbb{T}$ (but possibly not by the tropical variety). In particular, we study the relation between the Picard group of $X$ and $X_\mathbb{T}$. We solve the problem in the case when $X$ is a toric variety.
April 9 Chen Jiang
University of Tokyo
Noether inequality for 3-folds
For varieties of general type, it is natural to study the distribution of birational invariants and relations between invariants. We are interested in the relation between two fundamental birational invariants: the geometric genus and the canonical volume. For a minimal projective surface, M. Noether proved a fundamental inequality which is known as the Noether inequality. It is thus natural and important to study the higher dimensional analogue.
In these talks, I am going to present our recent progress in three dimensional Noether inequality. This is a joint work with Jungkai Chen and Meng Chen. We will show that the same type inequality holds for all projective 3-folds of general type with geometric genus greater than 20. Our inequality is optimal due to known examples found by M. Kobayashi in 1992. This proves that the optimal Noether inequality holds for all but finitely many families of projective 3-folds (up to deformation and birational equivalence).
In the first talk, we are going to work on general theory, and examples. The proof of almost all cases will be explained.
In the second talk, we will explain the most technical case of (1,2) surface fibrations. We will see how connectedness lemma and alpha-invariants are involed with birational geometry of general type varieties.
April 16 Chi Li
Purdue
On some minimization problems in K-stability
I will discuss minimization problems of normalized volume and CM weight in the study of K-stability of Fano varieties. The study of both problem depends heavily on MMP techniques. This talk is mostly based on joint works with Chenyang Xu and Xiaowei Wang.
April 30 Igor Krylov
Korea Institute for Advanced Study
Parameter spaces of del Pezzo surfaces and birational geometry of del Pezzo fibrations
Del Pezzo fibrations are one of the types of the Mori Fiber Space output of the MMP. There may be many models for the del Pezzo fibration and we would like to work with the best one. For example it is known that for conic bundles there exists a model with a smooth total space. I will describe a construction of parameter space of del Pezzo surfaces of degree 1 and 2 together with a notion of stability. Then I define what are the best models of del Pezzo fibrations of degrees 1 and 2 and show the existence of a good birational model.