Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem

We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.

[1]  Jakub Przybylo,et al.  Can Colour-Blind Distinguish Colour Palettes? , 2013, Electron. J. Comb..

[2]  Grigory Mikhalkin,et al.  Tropical curves, their Jacobians and Theta functions , 2006 .

[3]  Matthew Baker,et al.  Specialization of linear systems from curves to graphs , 2007 .

[4]  Sam Payne,et al.  The tropicalization of the moduli space of curves , 2012, 1604.03176.

[5]  D. Eisenbud,et al.  A simpler proof of the Gieseker-Petri theorem on special divisors , 1983 .

[6]  Serguei Norine,et al.  Riemann–Roch and Abel–Jacobi theory on a finite graph , 2006, math/0608360.

[7]  L. Narici,et al.  On Non‐Archimedean Analysis a , 1990 .

[8]  Vyassa L. Baratham,et al.  Towards a tropical proof of the Gieseker–Petri Theorem , 2012, 1205.3987.

[9]  Sam Payne,et al.  Lifting Divisors on a Generic Chain of Loops , 2014, Canadian Mathematical Bulletin.

[10]  D. Eisenbud,et al.  Limit linear series: Basic theory , 1986 .

[11]  Chang Mou Lim,et al.  A note on Brill-Noether thoery and rank determining sets for metric graphs , 2011, 1106.5519.

[12]  Amaury Thuillier.  Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d'Arakelov , 2005 .

[13]  M. Bigas,et al.  Petri map for vector bundles near good bundles , 2012, Journal of Pure and Applied Algebra.

[14]  Ye Luo.  Rank-determining sets of metric graphs , 2011, J. Comb. Theory, Ser. A.

[15]  Yoav Len,et al.  The Brill–Noether rank of a tropical curve , 2012, 1209.6309.

[16]  Josephine Yu,et al.  Linear systems on tropical curves , 2009, 0909.3685.

[17]  Joe Harris,et al.  On the variety of special linear systems on a general algebraic curve , 1980 .

[18]  Sam Payne,et al.  Nonarchimedean geometry, tropicalization, and metrics on curves , 2011, 1104.0320.

[19]  Brian Osserman.  A simple characteristic-free proof of the Brill-Noether theorem , 2011, 1108.4967.

[20]  Dhar,et al.  Self-organized critical state of sandpile automaton models. , 1990, Physical review letters.

[21]  Greg Kuperberg,et al.  CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM , 2013, Forum of Mathematics, Sigma.

[22]  R. Lazarsfeld.  Brill-Noether-Petri without degenerations , 1986 .

[23]  Michael Kerber,et al.  A Riemann–Roch theorem in tropical geometry , 2006, math/0612129.

[24]  D. Gieseker.  Stable curves and special divisors: Petri's conjecture , 1982 .

[25]  Rohit Agrawal,et al.  Involutions on Standard Young Tableaux and Divisors on Metric Graphs , 2013, Electron. J. Comb..

[26]  Sam Payne,et al.  A tropical proof of the Brill-Noether Theorem , 2010, 1001.2774.

[27]  Omid Amini,et al.  Linear series on metrized complexes of algebraic curves , 2012, 1204.3508.

[28]  Nathan Pflueger,et al.  Tropical Curves , 2015 .

[29]  P. Griffiths,et al.  Geometry of algebraic curves , 1985 .