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.

A greatest common divisor and a least common multiple of solutions of a linear matrix equation.

Abstract: We describe the explicit form of a left greatest common divisor and a least common multiple of solutions of a solvable linear matrix equation over a commutative elementary divisor domain. We prove that these left greatest common divisor and least common multiple are also solutions of the same equation.

Correspondences of a K3 surface with itself via moduli of sheaves. I

Abstract: Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$ Let $N(X)$ be Picard lattice of $X$ In mathAG/0309348 and mathAG/0606289, all divisors in moduli of $(X,H)$ (i e pairs $H\in N(X)$ with $\rk N(X)=2$) implying $M_X(v)\cong X$ were described They give some Mukai's correspondences of $X$ with itself Applying these results, we show that there exists $v$ and a codimension 2 submoduli in moduli of $(X,H)$ (i e a pair $H\in N(X)$ with $\rk N(X)=3$) implying $M_X(v)\cong X$, but this submoduli cannot be extended to a divisor in moduli with the same property There are plenty of similar examples We discuss the general problem of description of all similar submoduli and defined by them Mukai's correspondences of $X$ with itself and their compositions, trying to outline a possible general theory

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.

On Jacobian group arithmetic for typical divisors on curves

Abstract: In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves, improving on similar results by other authors. At that time, we could only state that a generic divisor was typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field. This article pins down an explicit characterization of these typical divisors, for an arbitrary smooth projective curve of genus g >= 1 having at least one rational point. We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove not only that the algorithms are correct if various divisors are typical, but also that the success of our algorithms provides a guarantee that the resulting output is correct and that the resulting input and/or output divisors are also typical. These results apply in particular to our earlier algorithms for C_{3,4} curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for C_{3,4} curves.