Topic
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.
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TL;DR: In this article, it was shown that any divisor n of q-1, where q is a power of p, belongs to the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0.
Abstract: A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.
3 citations
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TL;DR: In this article, the Poincar\'e problem for plane foliations with only one dicritical divisor was solved and an algorithm for computing a rational first integral of prefixed genus was given.
Abstract: We solve the Poincar\'e problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give an algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus $g
eq 1$ of any type of plane foliation $\cf$. When the number of dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends on suitable families of invariant curves. When dic$(\cf) = 2$, it proves that the degree of the rational first integral can be bounded only in terms of $g$, the degree of $\cf$ and the local analytic type of the dicritical singularities of $\cf$.
3 citations
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TL;DR: In this article, it was shown that for a canonical weak weak Fano-Fano (Fano $3$-fold) anti-canonical divisor, Cartier, Nef and big, there exists a terminal weak (i.e., Fano Fano)-Fano-Cartier (cf.
Abstract: By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical weak $\mathbb{Q}$-Fano $3$-fold $V$, we show that there exists a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, being birational to $V$, such that the $m$-th anti-canonical map defined by $|-mK_{X}|$ is birational for all $m\geq 52$. As an intermediate result, we show that for any $K$-Mori fiber space $Y$ of a canonical weak $\mathbb{Q}$-Fano $3$-fold, the $m$-th anti-canonical map defined by $|-mK_{Y}|$ is birational for all $m\geq 52$.
3 citations
TL;DR: In this paper, the authors describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition.
Abstract: We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor due to Hashimoto.
3 citations
TL;DR: In this article, the authors studied the number of divisors of a regular quadratic form with integral variables on average, and established an asymptotic formula of the sum of these divisores.
Abstract: Let f(x1, . . . , xs) (s ≥ 3) be a regular quadratic form with integral variables. We study the number of divisors of f(x1, . . . , xs) on average. We establish an asymptotic formula of the sum of these divisors.
3 citations