An explicit resolution of the Abel map via tropical geometry / Bertini’s theorem in positive characteristic

Abstract: An explicit resolution of the Abel map via tropical geometry, by Sally Andria (UFF)

In this talk I will talk about the problem studied in my thesis: Is it possible to find an explicit resolution of the Abel map (a rational map) for a nodal curve? We start with a family of curves, that is a regular smoothing of a nodal curve with smooth components. We take a polarization, an invertible sheaf, and a section through the smooth locus of the family. The Abel map is the rational map taking a tuple of points $(Q_1,\ldots,Q_d)$ on a curve of the family to the associated sheaf in the Esteves compactified Jacobian. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. The solution of the combinatorial problem gives rise to an explicit resolution of the Abel map.

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Bertini’s theorem in positive characteristic, by Cesar Hilario (IMPA)

A classical theorem of Bertini states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth, that is, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, Bertini’s theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating such a 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 $p \le 2g + 1$. When $g = 1$ and $g = 2$, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Cañate. A birational classification of the case $g = 3$ was started by Stohr ($p = 7, 5$), and then continued by Salomão ($p = 3$). In this talk I shall report on some progress in the case $g = 3$, $p = 2$. In fact, several examples show already that in this setting very interesting geometric phenomena arise.

algebraic geometry

Audience: researchers in the topic

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Brazilian algebraic geometry seminar

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Previous talks available at the YouTube channel "Brazilian Algebraic Geometry" www.youtube.com/channel/UCM-pcdNdpWxQFgOg-illE2w

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