It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic m× n matrix. In this paper we discuss the relations of t-minors for t < min(m,n). We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is...

Let Z ⊂ P, r ≥ 4, be a closed subscheme with dim(Z) ≤ r−4. Fix integers c > 0 and gi ≥ 0, i = 1, . . . , c. We prove that the general union of Z and c smooth curves Yi ⊂ P with genus gi and deg(Yi) ≥ r + gr as maximal rank (i.e. the expected postulation) if deg(Y1) + · · ·+ deg(Yc) 0.

We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ Degree-Maximal ⇒ Shifted which are equivalent concepts for 2-families (= simple graphs), but strict implications for kfamilies with k ≥ 3. The implication that uniquel...

Abstract We study subfields of quotient division algebras of domains of finite GK dimension and introduce a combinatorial property we call the straightening property. We show that many classes of algebras have this straightening property and show that if A is a domain of GK dimension d with this property that is not PI, then the maximal subfields of the quotient division algebra of A have trans...