arXiv subjects

Magnus Dehli Vigeland

Publications and source records attributed to Magnus Dehli Vigeland.

4 recordsLinked to original sources

arXiv2019

Tropical Lines on Cubic Surfaces

Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines.

math.AG↗

arXiv2007

Smooth tropical surfaces with infinitely many tropical lines

We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

math.AG↗

arXiv2007

Tropical complete intersection curves

A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected.

math.AG↗

arXiv2004

The group law on a tropical elliptic curve

In analogy with the classical group law on a plane cubic curve, we define a group law on a smooth plane tropical cubic curve. We show that the resulting group is isomorphic to $S^1$.

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