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Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines.

Algebraic Geometry

Speaker: Maria Angelica Cueto, Ohio State University
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Location: 2112 MSB
Start time: Wed, Oct 11 2017, 11:00AM

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-known statement ``any smooth surface of degree 3 in P^3 contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology. The novel idea is to consider the embedding of a smooth cubic surface in P^44 via its anticanonical bundle. The tropicalization induced by this embedding contains exactly 27 lines under a mild genericity assumption. More precisely, smooth cubic surfaces in P^3 are del Pezzos, and can be obtained by blowing up P^2 at six points in general position. We identify these points with six parameters over a field with nontrivial valuation. Our genericity assumption involves the valuations of 36 linear expressions in these parameters which give the positive roots of type E_6. Tropical convexity plays a central role in ruling out the existence of extra tropical lines on the tropical cubic surface.

This talk is based on an ongoing project joint with Anand Deopurkar.