284 research outputs found
Amoebas of algebraic varieties and tropical geometry
This survey consists of two parts. Part 1 is devoted to amoebas. These are
images of algebraic subvarieties in the complex torus under the logarithmic
moment map. The amoebas have essentially piecewise-linear shape if viewed at
large. Furthermore, they degenerate to certain piecewise-linear objects called
tropical varieties whose behavior is governed by algebraic geometry over the
so-called tropical semifield. Geometric aspects of tropical algebraic geometry
are the content of Part 2. We pay special attention to tropical curves. Both
parts also include relevant applications of the theories. Part 1 of this survey
is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in
Geometry
Geometry of tropical moduli spaces and linkage of graphs
We prove the following "linkage" theorem: two p-regular graphs of the same
genus can be obtained from one another by a finite alternating sequence of
one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the
linkage theorem to prove that various moduli spaces of tropical curves are
connected through codimension one.Comment: Final version incorporating the referees correction
Quantum indices and refined enumeration of real plane curves
We associate a half-integer number, called {\em the quantum index}, to
algebraic curves in the real plane satisfying to certain conditions. The area
encompassed by the logarithmic image of such curves is equal to times
the quantum index of the curve and thus has a discrete spectrum of values. We
use the quantum index to refine real enumerative geometry in a way consistent
with the Block-G\"ottsche invariants from tropical enumerative geometry.Comment: Version 4: exposition improvement, particularly in the proof of
Theorem 5 (following referee suggestions
Decomposition into pairs-of-pants for complex algebraic hypersurfaces
It is well-known that a Riemann surface can be decomposed into the so-called
pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3
points. We show that a smooth complex projective hypersurface of arbitrary
dimension admits a similar decomposition. The n-dimensional pair-of-pants is
diffeomorphic to the complex projective n-space minus n+2 hyperplanes.
Alternatively, these decompositions can be treated as certain fibrations on
the hypersurfaces. We show that there exists a singular fibration on the
hypersurface with an n-dimensional polyhedral complex as its base and a real
n-torus as its fiber. The base accomodates the geometric genus of a
hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n.Comment: 35 pages, 9 figures, final version to appear in Topolog
Counting curves via lattice paths in polygons
This note presents a formula for the enumerative invariants of arbitrary
genus in toric surfaces. The formula computes the number of curves of a given
genus through a collection of generic points in the surface. The answer is
given in terms of certain lattice paths in the relevant Newton polygon.
If the toric surface is the projective plane or the product of two projective
lines then the invariants under consideration coincide with the Gromov-Witten
invariants. The formula gives a new count even in these cases, where other
computational technique is available.Comment: The version to appear as the English part of a paper in C. R. Acad.
Sci. Pari
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