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 paper, it was shown that if m = n-3 then every Sm-invariant F-nef divisor on the moduli space of stable n-pointed curves of genus zero is linearly equivalent to a combination of boundary divisors.
3 citations
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TL;DR: In this paper, it was shown that for a point $P$ on the ramification curve of $\varphi$ at most sixteen exceptional curves go through in characteristic $2, and at most ten in all other characteristics.
Abstract: Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $\varphi$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that is ramified over a smooth curve of degree 6. The surface $X$ contains 240 curves with negative self-intersection, called exceptional curves. We prove that for a point~$P$ on the ramification curve of $\varphi$, at most sixteen exceptional curves go through~$P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, we prove that for a point $Q$ outside the ramification curve of $\varphi$, at most twelve exceptional curves go through $Q$ in characteristic $3$, and at most ten in all other characteristics. We show that these upper bounds are sharp in all cases except possibly in characteristic 5 outside the ramification curve.
3 citations
01 Aug 2002
TL;DR: In this article, a convolution-type formula for the number of partitions of n that are not divisible by r and coprime to r was given. And another result was given for the sum of the odd divisors of n.
Abstract: Let n;r be natural numbers, with r‚ 2. We present convolution-type formulas for the number of partitions of n that are (1) not divisible by r; (2) coprime to r. Another result obtained is a formula for the sum of the odd divisors of n.
3 citations
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TL;DR: A new approach to divisor scalar multiplication in Jacobian of genus 2 hyperelliptic curves over fields with odd characteristic, without field inversion is proposed.
Abstract: in this paper we proposed a new approach to divisor scalar multiplication in Jacobian of genus 2 hyperelliptic curves over fields with odd characteristic, without field inversion. It is based on improved addition formulae of the weight 2 divisors in projective divisor representation in most frequent case that suit very well to scalar multiplication algorithms based on Euclidean addition chains. Keywords-hyperelliptic curve, divisor, Jacobian, addition formulae, exponentiation, projective representation
3 citations
TL;DR: In this paper, Erdős and Wolke obtained upper bounds of the expected order of magnitude for sums of the form (sum n \le x) a_n \tau _r(n), where τ is the f-fold divisor function.
Abstract: Let \(\{a_n\}\) be a sequence of nonnegative real numbers. Under very mild hypotheses, we obtain upper bounds of the expected order of magnitude for sums of the form \(\sum _{n \le x} a_n \tau _r(n)\), where \(\tau _r(n)\) is the \(r\)-fold divisor function. This sharpens previous estimates of Friedlander and Iwaniec. The proof uses combinatorial ideas of Erdős and Wolke.
3 citations