333 research outputs found
A bit of tropical geometry
This friendly introduction to tropical geometry is meant to be accessible to
first year students in mathematics. The topics discussed here are basic
tropical algebra, tropical plane curves, some tropical intersections, and
Viro's patchworking. Each definition is explained with concrete examples and
illustrations. To a great exten, this text is an updated of a translation from
a french text by the first author. There is also a newly added section
highlighting new developments and perspectives on tropical geometry. In
addition, the final section provides an extensive list of references on the
subject.Comment: 27 pages, 19 figure
The separating semigroup of a real curve
We introduce the separating semigroup of a real algebraic curve of dividing
type. The elements of this semigroup record the possible degrees of the
covering maps obtained by restricting separating morphisms to the real part of
the curve. We also introduce the hyperbolic semigroup which consists of
elements of the separating semigroup arising from morphisms which are
compositions of a linear projection with an embedding of the curve to some
projective space.
We completely determine both semigroups in the case of maximal curves. We
also prove that any embedding of a real curve to projective space of
sufficiently high degree is hyperbolic. Using these semigroups we show that the
hyperbolicity locus of an embedded curve is in general not connected.Comment: 14 pages, 4 figures, published version, comments welcome
Brief introduction to tropical geometry
The paper consists of lecture notes for a mini-course given by the authors at
the G\"okova Geometry \& Topology conference in May 2014. We start the
exposition with tropical curves in the plane and their applications to problems
in classical enumerative geometry, and continue with a look at more general
tropical varieties and their homology theories.Comment: 75 pages, 37 figures, many examples and exercise
Non-existence of torically maximal hypersurfaces
Torically maximal curves (known also as simple Harnack curves) are real
algebraic curves in the projective plane such that their logarithmic Gau{\ss}
map is totally real. In this paper we show that hyperplanes in projective
spaces are the only torically maximal hypersurfaces of higher dimensions.Comment: 10 pages. V2 merges the first version of this paper with the first
version of arXiv:1510.0026
Toric degenerations of Grassmannians from matching fields
We study the algebraic combinatorics of monomial degenerations of Pl\"ucker
forms which is governed by matching fields in the sense of Sturmfels and
Zelevinsky. We provide a necessary condition for a matching field to yield a
Khovanskii basis of the Pl\"ucker algebra for -planes in -space. When the
ideal associated to the matching field is quadratically generated this
condition is both necessary and sufficient. Finally, we describe a family of
matching fields, called -block diagonal, whose ideals are quadratically
generated. These matching fields produce a new family of toric degenerations of
\Gr(3, n)
Tropicalization of del Pezzo surfaces
We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces
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