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: An efficiency analysis shows that the degenerate divisor as a base element can be a valid alternative in hyperelliptic curve cryptosystems as well.
1 citations
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TL;DR: In this paper, the authors generalized the Fujita approximation theorem to the case of multigraded linear series and showed that the volume of a big divisor on a projective variety can be approximated arbitrarily closely by the self-intersection number of the ample divisors on a birational modification of the same variety.
Abstract: The original Fujita approximation theorem states that the volume of a big divisor $D$ on a projective variety $X$ can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of $X$. One can also formulate it in terms of graded linear series as follows: let $W_{\bullet} = \{W_k \}$ be the complete graded linear series associated to a big divisor $D$: \[
W_k = H^0\big(X,\mathcal{O}_X(kD)\big). \] For each fixed positive integer $p$, define $W^{(p)}_{\bullet}$ to be the graded linear subseries of $W_{\bullet}$ generated by $W_p$: \[
W^{(p)}_{m}={cases}
0, &\text{if $p
mid m$;}
\mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.}
{cases} \] Then the volume of $W^{(p)}_{\bullet}$ approaches the volume of $W_{\bullet}$ as $p\to\infty$. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.
1 citations
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TL;DR: In this paper, the authors studied the period map for a family of $K3$ surfaces which is given by the anticanonial divisor of a toric variety and determined the period differential equation and its monodromy group.
Abstract: In this article we study the period map for a family of $K3$ surfaces which is given by the anticanonial divisor of a toric variety. We determine the period differential equation and its monodromy group. Moreover we show the exact relation between our period differential equation and the unifomizing differential equation of the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$.
1 citations
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TL;DR: In this paper, the authors provided a criterion for the very ampleness of a line bundle on a polarized abelian surface, where the line bundle is defined as the blow-up of a point at general points with exceptional divisors.
Abstract: Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot \pi^*L_S - \alpha \cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $\pi:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$ is an arbitrary surface of Picard number one with a numerically trivial canonical divisor.
1 citations
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TL;DR: In particular, this paper showed that the EGPS conjecture holds for infinite sets with counting function O(x^{\frac12 + \epsilon(x)}) and showed that for any positive numbers, there are positive numbers with arbitrarily many preimages lying between α(1-α) and β(1+α(α) n.
Abstract: Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d
1 citations