Bertini's theorem in positive characteristic

Abstract: The Bertini-Sard theorem is a classical result in algebraic geometry. It states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth; in other words, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, the Bertini-Sard theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating this failure naturally leads to the classification of its exceptions. By a theorem of Tate, a fibration by singular curves of arithmetic genus g in characteristic p > 0 may exist only if p <= 2g + 1. When g = 1 and g = 2, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Canate. A birational classification of the case g = 3 was started by Stohr (p = 7, 5), and then continued by Salomao (p = 3). In this talk I will report on some progress in the case g = 3, p = 2. In fact, a great variety of examples exist and very interesting geometric phenomena arise from them.

algebraic geometry

Audience: researchers in the discipline

American Graduate Student Algebraic Geometry Seminar

Series comments: The American Graduate Student Algebraic Geometry Seminar (AGSAGS) is a virtual seminar by and for algebraic geometry graduate students.

The goal of this seminar is for graduate students to share their research through online talks and to provide an algebraic geometry graduate networking system. Grad students, postdocs, and professors are welcome to attend.

Seminars will be held on Mondays at 4 p.m. Eastern on Zoom. We hope this time is convenient for graduate students in the Americas, hence the name AGSAGS. Prior registration is required and interested participants should register here: sites.google.com/view/agsags/registration. In addition to graduate talks, there will be occasional social events.