Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Algebraic-geometry codes, one-point codes, and evaluation codes

Abstract: One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed---Muller codes. Given an $${\mathbb{F}}_q$$ -algebra A, an order function $$\rho$$ on A and given a surjective $${\mathbb{F}}_q$$ -morphism of algebras $$\varphi: A\rightarrow {\mathbb{F}}_q^{n}$$ , the ith evaluation code with respect to $$A,\rho,\varphi$$ is defined as the code $$C_i=\varphi(\{f\in A: \rho(f)\leq i\})$$ . In this work it is shown that under a certain hypothesis on the $$\mathbb{F}_q$$ -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over $$\mathbb{F}_q$$ and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G i with associated divisors $$\Gamma_i$$ is the sequence of evaluation codes associated to some $${\mathbb{F}}_q$$ -algebra A, some order function $$\rho$$ and some surjective morphism $$\varphi$$ with $$\{f\in A: \rho(f)\leq i\}={\mathcal{L}}(\Gamma_i)$$ if and only if it is a sequence of one-point codes.

Surfaces close to the Severi lines in positive characteristic

Abstract: Let $X$ be a surface of general type with maximal Albanese dimension over an algebraically closed field of characteristic greater than two: we prove that if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. Moreover we give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. In addition we expose a similar partial result over algebraically closed fields of characteristic two. We also prove, in the same hypothesis, that a surface $X$ with $K_X^2 eq 4\chi(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq 4\chi(\mathcal{O}_X)+8(q-2)$ and we give a characterization of surfaces for which the equality holds. These are surfaces whose canonical model is a double cover of an isotrivial smooth elliptic surface branched over an ample divisor with at most negligible singularities whose intersection with the elliptic fibre is $4$.

Binary integer division processing method

Abstract: PURPOSE:To execute 63-bit integer type division without using a floating decimal point instruction by executing sorting by means of the number of the effective digits of divisors, in the case of the divisor at >= n bits, shifting a divident and the divisor, obtaining a value whose number of the effective digits is <=n bits, and obtaining a quoitient CONSTITUTION:A computer 10 has plural registers respectively holding data of n bits, and holds a binary integer division instruction executing mechanism set in the registers When a divident X is divided by a divisor Y so that the quoitient may be expressed in the range of 2n bits by the computer 10 whose one word consists of n bits, first the number of the effective digits is obtained By the number of the effective digits, a division error or a clear quoitient can be obtained Further by paying attention to the number of the effective digits of the divisor Y, sorting is executed, the division instruction by means of the divisor of n bitz is used, and the division is executed Thus the 63-bit integer type division in a FORTRAN program, etc, can be executed without using the floating decimal point instruction

A note on the projective varieties of almost general type

Abstract: A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial canonical divisors, under the minimal model conjecture. Furthermore we prove that, for a projective variety $X$ with only terminal singularities, $X$ is of almost general type if and only if the canonical divisor $K_X$ is almost nup, under the minimal model conjecture.