In [2], Billera proved that the R-algebra of continuous piecewise polynomial functions (C0 splines) on a d-dimensional simplicial complex 1 embedded in Rd is a quotient of the Stanley–Reisner ring A1 of 1. We derive a criterion to determine which elements of the Stanley–Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C k (1)...

then V is a primitive Fano variety of dimensionM , that is, Pic V = ZKV and (−KV ) is ample. The purpose of this note is to sketch a proof of the following Theorem 1. A general (in the sense of Zariski topology) variety V is birationally superrigid. In particular, V admits no non-trivial structures of a rationally connected fibration, any birational map V 99K V ♯ onto a Fano variety with Q-fact...

variety). For each i, let φi(x) be a C-dense quantifier-free Li-formula with parameters from K. Then we can find a K-definable rational function f : C → P which is non-constant, and has the property that the divisor f−1(0) is a sum of distinct points in ⋂n i=1 φi(K), with no multipliticities. (In particular, the support of the divisor contains no points from C(K)\C(K) and no points from C \ C.)...

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski’s predimension arises in this general context as a natural geometric notion and use it ...

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the...