January 11-13 (Friday-Sunday) |
Complex Algebraic Geometry 2019 takes place at UCSD. |
March 20 at 2-3pm in 7218 Maksym Fedorchuk (Boston College, note the unusual time) |
Standard models of low degree del Pezzo fibrations |
A del Pezzo fibration is one of the natural outputs of the Minimal Model Program for threefolds. At the same time, geometry of an arbitrary del Pezzo fibration can be unsatisfying due to the presence of non-integral fibers and terminal singularities of an arbitrarily large index. In 1996, Corti developed a program of constructing ‘standard models’ of del Pezzo fibrations within a fixed birational equivalence class. Standard models enjoy a variety of desired properties, one of which is that all of their fibers are ℚ-Gorenstein integral del Pezzo surfaces. Corti proved the existence of standard models for del Pezzo fibrations of degree d ≥ 2, with the case of d = 2 being the most difficult. The case of d = 1 remained a conjecture. In 1997, Kollár recast and improved the Corti’s result in degree d = 3 using ideas from the Geometric Invariant Theory for cubic surfaces. I will present a generalization of Kollár’s approach in which we develop notions of stability for families of low degree (d ≤ 2) del Pezzo fibrations in terms of their Hilbert points (i.e., low degree equations cutting out del Pezzos). A correct choice of stability and a bit of enumerative geometry then leads to (very good) standard models in the sense of Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov. |
May 3 at 4-5pm in 5829 Claudiu Raicu (Notre Dame) |
Koszul Modules and Green's Conjecture |
Formulated in 1984, Green's Conjecture predicts that one can recognize the intrinsic complexity of an algebraic curve from the syzygies of its canonical embedding. Green's Conjecture for a general curve has been resolved using geometric methods in two landmark papers by Voisin in the early 00s. I will explain how the theory of Koszul modules provides an alternative solution to this problem, by relating it via Hermite reciprocity to the study of the syzygies of the tangent developable surface to a rational normal curve. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman. |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018. Contact Jonathan Conder at jconder@ucsd.edu about problems with the website or posters.