Tropicalisation of rational varieties

A. DICKENSTEIN

Olinda, Brasil

Congreso; Pan-American Advanced Studies Institute in Commutative Algebra and its connections to Geometry; 2009

Universidad de Recife, Brasil

The tropicalisation trop(V) of an affine or projective algebraic variety V is a polyhedral complex, together with intersection theoretic data, that encodes important information about V as its dimension, its degree, and its asymptotic directions. When V is a hypersurface, trop(V) carries the same information as the Newton  polytope of a defining equation for V.The problem of describing trop(V) for a rational variety in terms of a given rational parametrization has been recently studied by Sturmfels-Tevelev-Yu and Esterov-Khovanskii. Based on Kapranov´s theorem, we present a naive approach to the problem using curve valuations, which allows us to slightly generalize their results. The advantage of this point of view is that even if we get complete results only when the polynomials defining the parametrization are generic with respect to their Newton polytopes, the proofs can be extended to deal with more general cases. This is joint work in progress with Bernard Mourrain.