BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sally Andria / César Hilário (UFF / IMPA) DTSTART:20200930T183000Z DTEND:20200930T200000Z DTSTAMP:20260423T021729Z UID:BRAG/25 DESCRIPTION:Title: An explicit resolution of the Abel map via tropical geometry / Bertini’s t heorem in positive characteristic\nby Sally Andria / César Hilário ( UFF / IMPA) as part of Brazilian algebraic geometry seminar\n\n\nAbstract\ nAn explicit resolution of the Abel map via tropical geometry\, by Sally A ndria (UFF)\n\nIn this talk I will talk about the problem studied in my th esis: Is it possible to find an explicit resolution of the Abel map (a rat ional map) for a nodal curve?\nWe start with a family of curves\, that is a regular smoothing of a nodal curve with smooth components. We take a pol arization\, an invertible sheaf\, and a section through the smooth locus o f 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 th e Esteves compactified Jacobian. We translate this problem into an explici t combinatorial problem by means of tropical and toric geometry. The solut ion of the combinatorial problem gives rise to an explicit resolution of t he Abel map.\n\n\n-- xx -- xx --\n\nBertini’s theorem in positive charac teristic\, by Cesar Hilario (IMPA)\n\nA classical theorem of Bertini state s that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth\, that is\, there do no t exist fibrations by singular varieties with smooth total space. Unfortun ately\, Bertini’s theorem fails in positive characteristic\, as was firs t observed by Zariski in the 1940s. Investigating such a failure naturally leads to the classification of its exceptions. By a theorem of Tate\, a f ibration 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 fi brations have been studied by Queen\, Borges Neto\, Stohr and Simarra Cañ ate. 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 s hall report on some progress in the case $g = 3$\, $p = 2$. In fact\, seve ral examples show already that in this setting very interesting geometric phenomena arise.\n LOCATION:https://researchseminars.org/talk/BRAG/25/ END:VEVENT END:VCALENDAR