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SUMMARY:Sally Andria (Universidade Federal Fluminense)
DTSTART:20200821T140000Z
DTEND:20200821T150000Z
DTSTAMP:20260423T021009Z
UID:LAGARTOS/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LAGARTOS/4/"
 >Abel maps for nodal curves via tropical geometry</a>\nby Sally Andria (Un
 iversidade Federal Fluminense) as part of (LAGARTOS) Latin American Real a
 nd Tropical Geometry Seminar\n\n\nAbstract\nLet $\\pi\\colon \\mathcal{C}\
 \rightarrow B$ be a regular smoothing of a nodal curve with smooth compone
 nts and a section $\\sigma$  of $\\pi$ through its smooth locus. \nLet $\\
 mu$ and $\\mathcal{L}$ be a polarization and an invertible\nsheaf of degre
 e $k$ on $\\mathcal{C}/B$. The Abel map $\\alpha^{d}_{\\mathcal{L}}$ is th
 e rational map \n$\\alpha^{d}_{\\mathcal{L}}\\colon \\mathcal{C}^d \\dashr
 ightarrow \\overline{\\mathcal{J}}_{\\mu}^{\\sigma}$ taking a tuple \nof p
 oints $(Q_1\,\\dots\,Q_d)$ on a fiber $C_b$ of $\\pi$ to the sheaf $\\math
 cal{O}_{C_b}(Q_1+\\dots+Q_d-d\\sigma(b))\\otimes \\mathcal{L}|_{C_b}$. Her
 e $\\overline{\\mathcal{J}}_{\\mu}^{\\sigma}$ denotes Esteves compactified
  Jacobian.\nAn interesting question is to find an explicit resolution of t
 he map $\\alpha^{d}_{\\mathcal{L}}$.\nWe translate this problem into an ex
 plicit combinatorial problem by means of tropical  and toric geometry. The
  solution of the combinatorial problem gives rise to an explicit resolutio
 n of the Abel map. We are able to use this technique to construct all the 
 degree-$1$ Abel maps and give a resolution of the degree-$2$ Abel-Jacobi m
 ap.\n
LOCATION:https://researchseminars.org/talk/LAGARTOS/4/
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