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...

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

Gromoll and Meyer have represented a certain exotic 7-sphere Σ as a biquotient of the Lie group G = Sp(2). We show for a 2-parameter family of left invariant metrics on G that the induced metric on Σ has strictly positive sectional curvature at all points outside four subvarieties of codimension ≥ 1 which we describe explicitly.

In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and prove a restriction property of our adjoint ideal sheaves using characteristic p methods.

We extend results of our previous papers, on ordinary multiple points of curves [9], and on the computation of their conductor [8], to ordinary multiple subvarieties of codimension one.

H(X,C) = ⊕p+q=kH (X). A class α ∈ H(X,Q) is said to be a rational Hodge class if its image in H(X,C) belongs to H(X). As is well known, the classes which are Poincaré dual to irreducible algebraic subvarieties of codimension p of X are degree 2p Hodge classes. The Hodge conjecture asserts that any rational Hodge class is a combination with rational coefficients of such classes. In the case of a...

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications,...