In this paper we address the problem of understanding the gaps that may occur in the subresultant sequence of two polynomials. We define the gap subresultant varieties and prove that they are rational and have the expected dimension. We also give explicitly their corresponding prime ideals. © 2004 Elsevier Ltd. All rights reserved. MSC: 12Y05; 68W30; 13P10

In this article, a log del Pezzo surface of index two means a projective normal non-Gorenstein surface S such that (S, 0) is a log-terminal pair, the anti-canonical divisor −KS is ample and that 2KS is Cartier. The log del Pezzo surfaces of index two are shown to be constructed from data (X,E,∆) called fundamental triplets consisting of a non-singular rational surface X, a simple normal crossin...

This paper examines percolation questions in a deterministic setting. In particular, I consider R, the set of elements of Z2 with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result.

In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), highest common factor (hcf), or greatest commonmeasure (gcm), of two or more integers (when at least one of them is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2] This notion can be extended to polynomials, see ...

One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor. The Fundamental Theorem of Arithmetic states that every integer exceeding 1 can ...

I have one update from last time, and this is aimed more at the experts. Rob pointed out that there was no reason that we know that a Cartier divisor can be expressed as a difference (or quotient) of effective Cartier divisors. More precisely, a Cartier divisor can be described cohomologically as follows. Let X be a scheme. We have a sheaf O of invertible functions. There is another sheaf K tha...

We define a Deligne-Mumford stack XD,r which depends on a scheme X, an effective Cartier divisor D ⊂ X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M̄g,n(X,β) corresponding to relative stable maps.

From a rational convex polytope of dimension r ≥ 2 J.P. Hansen constructed an error correcting code of length n = (q−1)r over the finite field Fq. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the ...