A note on the toric Newton spectrum of a polynomial

We describe a class of toric varieties in the $N$-dimensional affine space which are minimally defined by no less than $N-2$ binomial equations.

In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the sin...

This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly complicated examples.

In this work, we discuss graph like image of curves under moment maps and their relation with the Newton polygon of the curve, which has applications to Lagrangian torus fibration of Calabi-Yau manifolds.

We give a definition of Newton non degeneracy independent of the system of generators defining the variety. This definition extends the notion of Newton non degeneracy to varieties that are not necessarily complete intersection. As in the previous definition of non-degeneracy for complete intersection varieties, it is shown that the varieties satisfying our definition can be resolved with a toric modification. Using tools of both toric and tropical geometry we describe th...

Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely understood through objects from geometric combinatorics. While accessible and understandable, the class of toric varieties is also rich enough to illustrate many properties of algebraic varieties. Toric varieties are also ubiquitous in applications of mathematics, from tensors to statistical models to geo...

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is the projective plane or the product of two projective lines then the invariants under consideration coincide with the Gromov-Witten invariants. The formula ...

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the Euclidean n-space and holomorphic curves with cert...

We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers $b_1$ and $b_2$, associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of sma...