D.13.5 realizationMatroids_lib

Library:

realizationMatroids.lib

Purpose:

Deciding Relative Realizability for Tropical Fan Curves in 2-Dimensional Matroidal Fans

Authors:

Anna Lena Winstel, [email protected]

Overview:

In tropical geometry, one question to ask is the following: given a one-dimensional balanced polyhedral fan C which is set theoretically contained in the tropicalization trop(Y) of an algebraic variety Y, does there exist a curve X in Y such that trop(X) = C? This equality of C and trop(X) denotes an equality of both, the fans trop(X) and C and their weights on the maximal cones. The relative realization space of C with respect to Y is the space of all algebraic curves in Y which tropicalize to C.

This library provides procedures deciding relative realizability for tropical fan curves, i.e. one-dimensional weighted balanced polyhedral fans, contained in two-dimensional matroidal fans trop(Y) where Y is a projective plane.

Notation:

If Y is a projective plane in (n-1)-dimensional projective space, we consider trop(Y) in R^n/<1>. Moreover, for the relative realization space of C with respect to Y we only consider algebraic curves of degree deg(C) in Y which tropicalize to C.

Procedures:

D.13.5.1 realizationDim		For a given tropical fan curve C in trop(Y), where Y = V(I) is a projective plane, this routine returns the dimension of the relative realization space of C with respect to Y, that is the space of all algebraic curves of degree deg(C) in Y which tropicalize to C. If the realization space is empty, the output is set to -1.
D.13.5.2 irrRealizationDim		This routine returns the dimension of the irreducible relative realization space of the tropical fan curve C with respect to Y = V(I), that is the space of all irreducible algebraic curves of degree deg(C) in Y which tropicalize to C. If the irreducible relative realization space is empty, the output is set to -1.
D.13.5.3 realizationDimPoly		If C is a tropical fan curve contained in the tropicalization trop(Y) of the projective plane Y = V(I) such that the relative realization space M of C is non-empty, this routine returns the tuple (dim(M),f) where f is an example of a homogeneous polynomial of degree deg(C) cutting out a curve X in Y which tropicalizes to C. If M is empty, the output is set to -1.