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Let be a del Pezzo surface of degree one over an algebraically closed field, and let be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free -graded resolution of over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.
The Hilbert scheme of points in contains an irreducible component which generically represents distinct points in . We show that when is at most , the Hilbert scheme is reducible if and only if and . In the simplest case of reducibility, the component is defined by a single explicit equation, which serves as a criterion for deciding whether a given ideal is a limit of distinct points.
To understand the components of the Hilbert scheme, we study the closed subschemes of which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most . In particular, we show that the scheme corresponding to the Hilbert function is the minimal reducible example.
We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that nonabelian free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman’s nonlocally-indicable right orderable groups.
We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra along with a distinguished element such that is a ribbon Hopf algebra, where and . The element is closely related to the topological “half-twist”, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that is a (topological) half-ribbon Hopf algebra, but that is not the standard ribbon element. For , we show that there is no half-ribbon element such that is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.
In 1956, Brauer showed that there is a partitioning of the -regular conjugacy classes of a group according to the -blocks of its irreducible characters with close connections to the block theoretical invariants. In a previous paper, the first explicit block splitting of regular classes for a family of groups was given for the 2-regular classes of the symmetric groups. Based on this work, the corresponding splitting problem is investigated here for the 2-regular classes of the alternating groups. As an application, an easy combinatorial formula for the elementary divisors of the Cartan matrix of the alternating groups at is deduced.