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, the authors studied the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(\mathbb{C}^*)^n).
Abstract: We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(\mathbb{C}^*)^n$. This is the proalgebraic toric-completion $X_{\mathbb{Q}}$ of $X$. The ramification over the invariant divisor and the singular invariant divisors of $X$ impose topological constraints on the automorphisms of $X_{\mathbb{Q}}$. Considering this proalgebraic space as the toric functor on the adelic complex plane multiplicative semigroup, we calculate its automorphic group. Moreover we show that its vector bundle category is the direct limit of the respective categories of the finite toric varieties coverings defining the proalgebraic toric-completion.
1 citations
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TL;DR: In this paper, the number of distinct integers that have no prime factor and a divisor in the multiplication table is estimated in terms of the order in which distinct integers are free of prime factors.
Abstract: We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table which are free of prime factors $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.
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TL;DR: In this paper, Lapkova reproved her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum of divisor sums.
Abstract: In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum $\sum_{n \leq x} d( n ) d (n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.
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TL;DR: A new trigonometrical definition of the GCD of two integers a, b : (1) gcd ( a , b ) = 1 π ∫ 0 π cos [ ( b − a ) x ] sin 2 ( a b x) sin ( a x ) sin ( b x ).
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TL;DR: In this paper, the authors deduce necessary conditions for the number s of singular fibers being 5, provided that the genus of the fibration is greater than or equal to 2 and is at least 6 if the surface is not birationally ruled.
Abstract: Let X be a non-singular, projective surface and $$f: X\rightarrow \mathbb {P}^1$$
a non-isotrivial, semistable fibration defined over $$\mathbb {C}$$
. It is known that the number s of singular fibers must be at least 5, provided that the genus of the fibration is greater than or equal to 2 and is at least 6 if the surface is not birationally ruled. In this paper, we deduce necessary conditions for the number s of singular fibers being 5. Concretely, we prove that if $$s=5$$
, then the condition $$(K_X+F)^2=0$$
holds unless S is rational and $$g\le 17$$
. The proof is based on a “vertical”version of Miyaoka’s inequality and positivity properties of the relative canonical divisor.
1 citations