669 research outputs found

    An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

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    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

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    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 CP3\mathbb C P^3

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    We establish a formula for the Gromov-Witten-Welschinger invariants of CP3\mathbb CP^3 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 CP3\mathbb CP^3 could be computed in terms of enumerative invariants of CP1×CP1\mathbb CP^1\times\mathbb CP^1 and of elliptic curves.Comment: 14 pages, 4 figures, minor corrections following referee's suggestion

    Cayley-Bacharach Formulas

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    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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