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# Copyright (c) 1997-2024
# Ewgenij Gawrilow, Michael Joswig, and the polymake team
# Technische Universität Berlin, Germany
# https://polymake.org
#
# This program is free software; you can redistribute it and/or modify it
# under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 2, or (at your option) any
# later version: http://www.gnu.org/licenses/gpl.txt.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#-------------------------------------------------------------------------------
object Curve {
# @category Weights and lattices
# The rows of this IncidenceMatrix correspond to the nodes of the tropical graph,
# the columns to the edges.
# Therefore, each row records the indices of the edges incident to that vertex.
property EDGES_THROUGH_VERTICES : IncidenceMatrix;
# @category Weights and lattices
# Some edges can be "marked", which means they go off to infinity, and may not
# be permuted by any automorphism.
# By default there are no marked edges.
property MARKED_EDGES : Set<Int>;
# @category Weights and lattices
# Each vertex may have an integer weight, which is affected by edge contractions:
# if a loop on that vertex is contracted, the weight increases by 1,
# if a non-loop edge incident to that vertex is contracted, the new weight
# is the sum of the weights of the two endpoints.
# Default weights are zero.
property VERTEX_WEIGHTS : Array<Int>;
# @category Weights and lattices
# Each edge may have a Scalar length.
# These are taken into account when determining isomorphism, but not when
# calculating a MODULI_CELL.
property EDGE_LENGTHS : Vector<Scalar>;
# @category Weights and lattices
# Some additional inequalities may be imposed on the lengths of edges.
# The columns of this Matrix correspond to the columns of EDGES_THROUGH_VERTICES.
property INEQUALITIES : Matrix<Scalar>;
# @category Weights and lattices
# The simplicial complex that records exactly one vertex for each
# isomorphism class of assignments of lengths to the edges.
# See the documentation of the function moduli_cell() for an example.
property MODULI_CELL : topaz::GeometricSimplicialComplex;
rule MODULI_CELL : EDGES_THROUGH_VERTICES {
$this->MODULI_CELL = moduli_cell($this);
}
# @category Combinatorics
# Number of edges of the underlying graph.
property N_EDGES : Int;
rule N_EDGES : EDGES_THROUGH_VERTICES {
$this->N_EDGES = $this->EDGES_THROUGH_VERTICES->cols();
}
weight 0.10;
# @category Combinatorics
# Number of vertices (or nodes) of the underlying graph.
property N_VERTICES : Int;
rule N_VERTICES : EDGES_THROUGH_VERTICES {
$this->N_VERTICES = $this->EDGES_THROUGH_VERTICES->rows();
}
weight 0.10;
# @category Weights and lattices
# Genus of the abstract tropical curve, taking [[VERTEX_WEIGHTS]] into account
# @example with weights explicitly given
# > $C = new Curve(EDGES_THROUGH_VERTICES=>[[0,1],[1]], VERTEX_WEIGHTS=>[0,1]);
# > print $C->GENUS;
# | 2
# @example genus as abstract graph
# > $K4 = new Curve(EDGES_THROUGH_VERTICES=>[[0,1,2],[0,3,4],[1,3,5],[2,4,5]]);
# > print $K4->GENUS;
# | 3
property GENUS : Int;
rule GENUS : N_EDGES, N_VERTICES, VERTEX_WEIGHTS {
$this->GENUS = $this->N_EDGES - $this->N_VERTICES + 1 + sum($this->VERTEX_WEIGHTS);
}
# default weights are zero
rule VERTEX_WEIGHTS : N_VERTICES {
$this->VERTEX_WEIGHTS = new Array<Int>($this->N_VERTICES);
}
weight 0.10;
# no marked edges by default
rule MARKED_EDGES : {
$this->MARKED_EDGES = new Set<Int>();
}
weight 0.10;
}
# @category Weights and lattices
# Construct the stacky fan corresponding to a Curve.
# This is a fan with one representative for each orbit of faces under
# the symmetry group of the graph.
# @param Curve G
# @return fan::DisjointStackyFan
# @example
# To find the stacky f-vector of the complex of tropically smooth K_4 plane curves,
# first construct the appropriate Curve:
# > $UnitMatrix = unit_matrix(6);
# > ($u, $v, $w, $x, $y, $z) = 0..5;
# > ($U, $V, $W, $X, $Y, $Z) = map { new Vector($UnitMatrix->[$_]) } (0..5);
# > $D_ineqs = new Matrix( [$U-$V, $V-$W, $W-$Y, $W-$Z, $V-$X] );
# > $skeleton = new IncidenceMatrix([[$u,$v,$x],[$v,$w,$z],[$u,$w,$y],[$x,$y,$z]]);
# > $g = new Curve(EDGES_THROUGH_VERTICES=>$skeleton, INEQUALITIES=>$D_ineqs);
# Now we are ready to compute the stacky f-vector:
# > print stacky_le_fan($g)->STACKY_F_VECTOR;
# | 7 18 25 21 8
user_function stacky_le_fan(Curve) {
my ($tg) = @_;
my $autos = auto_group_on_coordinates($tg);
my $n_marked_edges = $tg->MARKED_EDGES->size();
my $n_unmarked_edges = $tg->EDGES_THROUGH_VERTICES->cols - $n_marked_edges;
my $ineqs = new Matrix(unit_matrix($n_unmarked_edges));
if (defined(my $additional_ineqs = $tg->lookup("INEQUALITIES"))) {
$ineqs /= $additional_ineqs;
}
my $sf = ($autos->size() != 0)
? fan::stacky_le_fan(new polytope::Cone(INEQUALITIES=>$ineqs, GROUP=>new group::Group(HOMOGENEOUS_COORDINATE_ACTION=>new group::PermutationAction(GENERATORS=>$autos))))
: fan::stacky_le_fan(new polytope::Cone(INEQUALITIES=>$ineqs));
return $sf;
}
# @category Weights and lattices
# Calculate the moduli cell of a given tropical graph G.
# This is a simplicial complex with one vertex for each isomorphism class
# of an assignment of lengths to the edges of G, that triangulates the
# space of all such isomorphism classes.
# @param Curve G the input graph. EDGE_LENGTHS are ignored.
# @option Int verbosity 0 (default) = off ... 5 (most verbose)
# @return topaz::GeometricSimplicialComplex
# @example
# The following incidence matrix defines a triangle with edges x,y,z,
# along extra edges a,b,c incident to each vertex.
# > ($a,$b,$c,$x,$y,$z,$w)=0..6;
# > $etv = new IncidenceMatrix([[$a,$x,$z],[$b,$x,$y],[$c,$y,$z]]);
# We consider the edges a,b,c to be marked.
# > $g = new Curve(EDGES_THROUGH_VERTICES=>$etv, MARKED_EDGES=>[$a,$b,$c]);
# The resulting moduli cell is homeomorphic to a 2-sphere:
# > print $g->MODULI_CELL->HOMOLOGY;
# | ({} 0)
# | ({} 0)
# | ({} 1)
# You can also include a matrix of INEQUALITIES among the edges, but be careful
# because the columns of that matrix correspond to the _unmarked_ edges:
# > $h = new Curve(EDGES_THROUGH_VERTICES=>$etv, MARKED_EDGES=>[$a,$b,$c], INEQUALITIES=>new Matrix([[1,-1,0]]));
# > print $h->MODULI_CELL->HOMOLOGY;
# | ({} 0)
# | ({} 0)
# | ({} 0)
user_function moduli_cell(Curve, { verbosity=>0 }) {
my ($tg, $options) = @_;
return moduli_cell_of_curve(fan::stacky_fundamental_domain(stacky_le_fan($tg), $options), $tg, $options);
}
# Local Variables:
# mode: perl
# cperl-indent-level: 3
# indent-tabs-mode:nil
# End:
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