Tropical tangents for complete intersection curves

Nathan Ilten; Yoav Len

doi:10.1090/mcom/3782

Tropical tangents for complete intersection curves

Nathan Ilten, Yoav Len

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the tropicalization of tangent lines to a complete intersection curve X in ℙⁿ. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X^∗ and tangential variety τ(X) of X. In particular, we are able to compute the degrees of X^∗and τ(X) and the Newton polytope of τ(X) without using any elimination theory.

Original language	English
Pages (from-to)	931-979
Number of pages	49
Journal	Mathematics of Computation
Volume	92
Issue number	340
Early online date	26 Oct 2022
DOIs	https://doi.org/10.1090/mcom/3782
Publication status	Published - 1 Mar 2023

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Copyright © The Author(s) 2022. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1090/mcom/3782.
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https://arxiv.org/abs/2104.15059

Cite this

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title = "Tropical tangents for complete intersection curves",

abstract = "We consider the tropicalization of tangent lines to a complete intersection curve X in ℙn. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X∗ and tangential variety τ(X) of X. In particular, we are able to compute the degrees of X∗and τ(X) and the Newton polytope of τ(X) without using any elimination theory.",

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N2 - We consider the tropicalization of tangent lines to a complete intersection curve X in ℙn. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X∗ and tangential variety τ(X) of X. In particular, we are able to compute the degrees of X∗and τ(X) and the Newton polytope of τ(X) without using any elimination theory.

AB - We consider the tropicalization of tangent lines to a complete intersection curve X in ℙn. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X∗ and tangential variety τ(X) of X. In particular, we are able to compute the degrees of X∗and τ(X) and the Newton polytope of τ(X) without using any elimination theory.

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Tropical tangents for complete intersection curves

Abstract

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