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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 the intersection numbers of a principally polarized abelian variety are characterized by their intersection numbers with a fixed numerical class in the N\'eron-Severi group.
Abstract: To every abelian subvariety of a principally polarized abelian variety $(A, \mathcal{L})$ we canonically associate a numerical class in the N\'eron-Severi group of $A$. We prove that these classes are characterized by their intersection numbers with $ \mathcal{L}$; moreover, the cycle class induced by an abelian subvariety in the Chow ring of $A$ modulo algebraic equivalence can be described in terms of its numerical divisor class. Over the field of complex numbers, this correspondence gives way to an explicit description of the (coarse) moduli space that parametrizes non-simple principally polarized abelian varieties with a fixed numerical class.
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TL;DR: In this article, the authors considered the simplest quartic number fields, defined by the irreducible quartic polynomials, and established an asymptotic formula for the number of simple quartic fields with discriminant and given index.
Abstract: We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.
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TL;DR: In this article, a very ample line bundle on a smooth curve of genus g was generated for a smooth projective surface, where g is a triple covering of genus p curve and p is a divisor.
Abstract: Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2} \max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrel{\phi}\to C'$ and $D$ a divisor on $C'$ with $4p<\deg D< \frac{g-1}{6}-2p$. Then $K_C(-\phi^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.
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TL;DR: In this paper, an analogue of the generalized divisor function in a number field was studied, namely, the Dirichlet series associated to this function, and an expression for the Riesz sum associated to it was derived by using convergence theorems.
Abstract: The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $\sigma_{\mathbb{K},\alpha}(n)$. The Dirichlet series associated to this function is $\zeta_{\mathbb{K}}(s)\zeta_{\mathbb{K}}(s-\alpha)$. We give an expression for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $\alpha=0$, gives the Vorono\"i summation formula associated to the divisor counting function $d(n)$. We also obtain a big $O$-estimate for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n)$.
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TL;DR: In this article, two tau functions, $\tau$ and $\hat{\tau}$, were defined on moduli spaces of spectral covers of Hitchin's systems. And they were used to compute the divisor of canonical 1-forms with multiple zeros.
Abstract: We define two tau functions, $\tau$ and $\hat{\tau}$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $\tau$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hat{\tau}$ is used to compute the divisor of canonical 1-forms with multiple zeros.

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021157
2020172
2019127
2018120
2017140