669 research outputs found
An ultradiscrete integrable map arising from a pair of tropical elliptic pencils
We present a tropical geometric description of a piecewise linear map whose
invariant curve is a concave polygon. In contrast to convex polygons, a concave
one is not directly related to tropical geometry; nevertheless the description
is given in terms of the addition formula of a tropical elliptic curve. We show
that the map is arising from a pair of tropical elliptic pencils each member of
which is the invariant curve of the ultradiscrete QRT map.Comment: 14 pages, 4 figure
On the addition formula for the tropical Hesse pencil
We give the addition formula for the tropical Hesse pencil, which is the
tropicalization of the Hesse pencil parametrized by the level-three theta
functions, via those for the ultradiscrete theta functions. The ultradiscrete
theta functions are reduced from the level-three theta functions through the
procedure of ultradiscretization by choosing their parameters appropriately.
The parametrization of the level-three theta functions firstly introduced in
\cite{KKNT09} gives an explicit correspondence between the amoeba of the real
part of the Hesse cubic curve and the tropical Hesse curve. Moreover, through
the parametrization, we obtain the subtraction-free forms of the addition
formulae for the level-three theta functions, which lead to the addition
formula for the tropical Hesse pencil in terms of the ultradiscretization.
Using the parametrization, the tropical counterpart of the Hesse configuration
is also given.Comment: 21 pages, 4 figure
Pencils of quadrics and Gromov-Witten-Welschinger invariants of
We establish a formula for the Gromov-Witten-Welschinger invariants of
with mixed real and conjugate point constraints. The method is
based on a suggestion by J. Koll\'ar that, considering pencils of quadrics,
some real and complex enumerative invariants of could be
computed in terms of enumerative invariants of
and of elliptic curves.Comment: 14 pages, 4 figures, minor corrections following referee's
suggestion
Cayley-Bacharach Formulas
The Cayley-Bacharach Theorem states that all cubic curves through eight given
points in the plane also pass through a unique ninth point. We write that point
as an explicit rational function in the other eight.Comment: 13 pages, 4 figure
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