HOPS: THEIR VARIETIES AND CULTIVATION

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...

Skolem Rings and Their Varieties

Prym varieties and their moduli

This survey discusses the geometry of the moduli space of Prym varieties. Several applications of Pryms in algebraic geometry are presented. The paper begins with with a historical discussion of the life and achievements of Friedrich Prym. Topics treated in subsequent sections include singularities and Kodaira dimension of the moduli space, syzygies of Prym-canonical embedding and the geometry ...

On Spinor Varieties and Their Secants⋆

We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type Dn, cubic equations exist if and only if n ≥ 9. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.

Stable Spherical Varieties and Their Moduli

We introduce a notion of stable spherical variety which includes the spherical varieties under a reductive group G and their flat equivariant degenerations. Given any projective space P where G acts linearly, we construct a moduli space for stable spherical varieties over P, that is, pairs (X, f), where X is a stable spherical variety and f : X → P is a finite equivariant morphism. This space i...