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 a normal projective variety and a non-isomorphic polarized endomorphism can be seen as a toric variety, then it is a Toric variety.
Abstract: Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety. Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-etale and $K_X+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-etale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.
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TL;DR: In this article, the asymptotic properties of σ−α(f(a(m))), where 0 < α ≤ 1, σ −α(n) = ∑ l|n 1 lα, f(x) be a polynomial with integer coefficients, were studied.
Abstract: For any positive integer m, let a(m) denotes the integer part of the k-th root of m. That is, a(m) = [ m ] . In this paper, we study the asymptotic properties of σ−α(f(a(m))), where 0 < α ≤ 1 be a fixed real number, σ−α(n) = ∑ l|n 1 lα , f(x) be a polynomial with integer coefficients. An asymptotic formula is obtained.
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TL;DR: In this paper, it was shown that the class of intersection of the divisor of schemes incident to a fixed line with any other class of a basis of the Chow ring $A^*(mathbb{P}^{2[N]})$ due to Mallavibarrena and Sols.
Abstract: We prove that there is an algorithm to compute the class of the intersection of the divisor of schemes incident to a fixed line with any other class of a basis of the Chow ring $A^*(\mathbb{P}^{2[N]})$ due to Mallavibarrena and Sols. This is progress towards a combinatorial description of the intersection product on the Hilbert scheme of points in the projective plane.
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TL;DR: In this article, the essential minimum of a normal and geometrically integral projective variety over a global field $K$ is shown to be the maximal slope of a Cartier divisor under mild positivity assumptions.
Abstract: Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb{R}}$ -Cartier divisor on $X$ . We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm{ess}}(\bar{D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm{ess}}(\bar{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = {\mathbb{P}}_K^{d}$ , our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Remond.
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01 Jan 2013
TL;DR: In this article, Kashiwara and Mochizuki showed that the Hermitian dual of a holonomic D-module is holonomic, generalizing the original result of M. Keshara for regular D-modules to possibly irregular H-D-modules.
Abstract: This chapter is similar to Chap. 10, but we now assume that D is a divisor with normal crossings. We start by proving the many-variable version of the Hukuhara–Turrittin theorem, that we have already encountered in the case of a smooth divisor. It will be instrumental for making the link between formal and holomorphic aspects of the theory. The new point in the proof of the Riemann–Hilbert correspondence is the presence of non-Hausdorff eale spaces, and we need to use the level structure to prove the local essential surjectivity of the Riemann–Hilbert functor. As an application of the Riemann–Hilbert correspondence in the good case and of the fundamental results of K. Kedlaya and T. Mochizuki on the elimination of turning points by complex blowing-ups, we prove a conjecture of M. Kashiwara asserting that the Hermitian dual of a holonomic \(\mathcal{D}\)-module is holonomic, generalizing the original result of M. Kashiwara for regular holonomic \(\mathcal{D}\)-modules to possibly irregular holonomic \(\mathcal{D}\)-modules and the result of Chap. 6 to higher dimensions.