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.

Generalization of the divisor products and proper divisor products sequences

Abstract: Let n be a positive integer, pd(n) denotes the product of all positive divisors of n, qd(n) denotes the product of all proper divisors of n. In this paper, we study the properties of the sequences of {pd(n)} and {qd(n)}, and prove that the generalized results for the sequences {pd(n)} and {qd(n)}.

Grossberg-Karshon twisted cubes and hesitant walk avoidance

Abstract: Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots, i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$ of the Weyl group of $G$, where the $s_i$ are the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to $\lambda$ and $\mathcal{I}$, which they called a twisted cube, whose lattice points encode (counted with sign according to a density function) characters of representations of $G$. In recent work, the first author and Jihyeon Yang prove that the Grossberg-Karshon twisted cube is untwisted (so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed from the data of $\lambda$ and $\mathcal{I}$, is basepoint-free. This corresponds to the situation in which the Grossberg-Karshon character formula is a true combinatorial formula in the sense that there are no terms appearing with a minus sign. In this note, we translate this toric-geometric condition to the combinatorics of $\mathcal{I}$ and $\lambda$. More precisely, we introduce the notion of hesitant $\lambda$-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is hesitant-$\lambda$-walk-avoiding.

Group theoretic characterizations of Buekenhout---Metz unitals in $\mathop{\mathrm{PG}}(2,q^{2})$

Abstract: Let G be the group of projectivities stabilizing a unital $\mathcal{U}$ in $\mathop{\mathrm{PG}}(2,q^{2})$ and let A,B be two distinct points of $\mathcal{U}$ . In this paper we prove that, if G has an elation group of order q with center A and a group of projectivities stabilizing both A and B of order a divisor of q?1 greater than $2(\sqrt{q}-1)$ , then $\mathcal{U}$ is an ovoidal Buekenhout---Metz unital. From this result two group theoretic characterizations of orthogonal Buekenhout---Metz unitals are given.

Brill-Noether loci for divisors on irregular varieties

Abstract: For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent space. If X is a smooth surface of maximal Albanese dimension and C is a curve on X, we define a Brill-Noether number \rho(C, r) and we prove, under some mild additional assumptions, that if \rho(C, r) is non negative then W^r(C,X) is nonempty of dimension bigger or equal to \rho(C,r). As an application, we derive lower bounds for h^0(K_D) for a divisor D that moves linearly on a smooth projective variety X of maximal Albanese dimension and inequalities for the numerical invariants of curves that do not move linearly on a surface of maximal Albanese dimension.

Parametrization of Kloosterman sets and $\mathrm{SL}_3$-Kloosterman sums

Abstract: We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $\Gamma_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.