Points in Generic Position and Conductor of Varieties with Ordinary Multiple Subvarieties of Codimension One

Ordinary Subvarieties of Codimension One

In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one.

Lifting Problem in Codimension 2 and Initial Ideals

Degrees of High-dimensional Subvarieties of Determinantal Varieties

Let n = pab, where p is a prime, and g.c.d.(p, b) = 1. In Pn −1, let Xr be the variety defined by rank((xi,j)) ≤ n − r. We show that any subvariety of Xr of codimension less than par must have degree a multiple of p. We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are

Lifting Tropical Intersections

We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proof...

Adelic Amoebas Disjoint from Open Halfspaces

We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropic...