Shifted varieties and discrete neighborhoods around varieties

Neighborhoods of Analytic Varieties in Complex Manifolds

On the Volume of Tubular Neighborhoods of Real Algebraic Varieties

The problem of determining the volume of a tubular neighborhood has a long and rich history. Bounds on the volume of neighborhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition numbers in numerical analysis. We present a self-contained derivation of bounds on the probability that a random point, chosen uniformly from a ball, lies within a...

Minimal Noncommutative Varieties and Power Varieties

A variety of finite monoids is a class of finite monoids closed under taking submonoids, quotients and finite direct products. A language L is a subset of a finitely generated free monoid. The variety theorem of Eilenberg sets up a one to one correspondence between varieties of finite monoids and classes of languages called, appropriately, varieties of languages. Recent work in variety theory h...

Secant Varieties of Toric Varieties

If X is a smooth projective toric variety of dimension n we give explicit conditions on characters of the torus giving an embedding X →֒ Pr that guarantee dimSecX = 2n+ 1. We also give necessary and sufficient conditions for a general point of SecX to lie on a unique secant line when X is embedded into Pr using a complete linear system. For X of dimension 2 and 3 we give formulas for deg SecX in...

Varieties of Algebras and their Congruence Varieties

1. Congruence lattices. G. Birkhoff and O. Frink noted that the congruence lattice Con04) of an algebra A (with operations of finite rank) is algebraic or compactly generated. The celebrated Grätzer-Schmidt theorem states that, conversely, every algebraic lattice is isomorphic to Con(A) for some algebra A. The importance of this is obvious, for it shows that unless something more is known about...