The tropics
We start with the simplest of the three worlds, the tropical world. Tropical
geometry is a kind of piecewise linear combinatorial geometry which arises when
one starts to think about algebraic geometry over the so-called tropical semi-ring.
This chapter will give a rather shallow introduction to the subject. We will start
with the definition of the tropical semi-ring and some elementary algebraic geometry
over the tropical semi-ring. We move on to the notion of parameterized tropical
curve, which features in Mikhalkin’s curve counting results. Next, we introduce the
type of tropical objects which arise in the Gross-Siebert program: affine manifolds
with singularities. These arise naturally if one wants to think about curve counting
in Calabi-Yau manifolds. We end with a duality between such objects given by the
Legendre transform.
1.1. Tropical hypersurfaces
We begin with the tropical semi-ring,
= (R, ⊕, ).
Here R is the set of real numbers, but with addition and multiplication defined by
a b := min(a, b)
a b := a + b.
Of course there is no additive inverse. This semi-ring became known as the tropical
semi-ring in honour of the Brazilian mathematician Imre Simon. The word tropical
has now spread rapidly.
We would like to do algebraic geometry over the tropical semi-ring instead of
over a field. Of course, since there is no additive identity in this semi-ring, it is not
immediately obvious what the zero-locus of a polynomial should be. The correct,
or rather, useful, intepretation is as follows. Let
. . . , xn]
denote the space of functions f :
R given by tropical polynomials
f(x1,...,xn) =
ai1,...,in x11
· · ·
where S
is a finite index set. Here all operations are in
so this is really
the function
f(x1,...,xn) = min{ai1,...,in +
ikxk | (i1,...,in) S}
This is a piecewise linear function, and the tropical hypersurface defined by f,
V (f)
as a set, is the locus where f is not linear.
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