Showing papers on "Divisor published in 2011"

Kahler-Einstein metrics with edge singularities

Abstract: This article considers the existence and regularity of Kahler-Einstein metrics on a compact Kahler manifold $M$ with edge singularities with cone angle $2\pi\beta$ along a smooth divisor $D$. We prove existence of such metrics with negative, zero and some positive cases for all cone angles $2\pi\beta\leq 2\pi$. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along $D$ for all $2\pi\beta < 2\pi$.

Existence of log canonical flips and a special LMMP

Abstract: Let $(X/Z,B+A)$ be a $\Q$-factorial dlt pair where $B,A\ge 0$ are $\Q$-divisors and $K_X+B+A\sim_\Q 0/Z$. We prove that any LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.

A canonical linear system associated to adjoint divisors in characteristic $p > 0$

Abstract: Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$. We further generalize this to incorporate a boundary divisor $\Delta$. We show that these subsystems behave like the global sections associated to multiplier ideals, $H^0(X, \mJ(X, \Delta) \tensor L)$ in characteristic zero. In particular, we show that these systems are in many cases base-point-free. While the original proof utilized Kawamata-Viehweg vanishing and variants of multiplier ideals, our proof uses test ideals.

Diffractive Geometric Optics for Bloch Wave Packets

Abstract: We study, for times of order \({1/\varepsilon}\), solutions of wave equations which are \({\fancyscript{O}(\varepsilon^2)}\) modulations of an \({\varepsilon}\) periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order \({\varepsilon}\). We construct accurate approximate solutions of the three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrodinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of the small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales \({t=\fancyscript{O}(1)}\). A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H1 asymptotics when the envelopes are only H1.

The cone of type A, level one conformal blocks divisors

Abstract: We prove that the type A, level one, conformal blocks divisors on $\bar{M}_{0,n}$ span a finitely generated, full-dimensional subcone of the nef cone. Each such divisor induces a morphism from $\bar{M}_{0,n}$, and we identify its image as a GIT quotient parameterizing configurations of points supported on a flat limit of Veronese curves. We show how scaling GIT linearizations gives geometric meaning to certain identities among conformal blocks divisor classes. This also gives modular interpretations, in the form of GIT constructions, to the images of the hyperelliptic and cyclic trigonal loci in $\bar{M}_{g}$ under an extended Torelli map.

Ramification and cleanliness

Abstract: This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell e p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth, separated and quasi-compact $k$-scheme, $D$ a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ a finite local ${\mathbb Z}_{\ell} $-algebra and ${\mathscr F}$ a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a {\em boundedness} condition on the ramification of ${\mathscr F}$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the {\em characteristic cycle} of ${\mathscr F}$. The cleanliness condition extends the one introduced by Kato for rank 1 sheaves. Roughly speaking, it means that the ramification of ${\mathscr F}$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for ${\mathscr F}$. Some cases of this formula have been previously proved by Kato and by the second author (T. S.).

The defect of Fano 3-folds

Abstract: This paper studies the rank of the divisor class group of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, there is a small modification of Y with a second extremal ray; Cutkosky, following Mori, gave an explicit geometric description of the contractions of extremal rays on terminal Gorenstein 3-folds. I introduce the category of weak-star Fanos, which allows one to run the Minimal Model Program (MMP) in the category of Gorenstein weak Fano 3-folds. If Y does not contain a plane, the rank of its divisor class group can be bounded by running a MMP on a weak-star Fano small modification of Y . These methods yield more precise bounds on the rank of ClY depending on the Weil divisors lying on Y . I then study in detail quartic 3-folds that contain a plane and give a general bound on the rank of the divisor class group of quartic 3-folds. Finally, I indicate how to bound the rank of the divisor class group of higher genus terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.

Explicit upper bounds for the remainder term in the divisor problem

Abstract: We first report on computations made using the GP/PARI package that show that the error term ∆(x) in the divisor problem is $= \mathscr{M} (x, 4) + O^* (0.35 x^{1/4} \log x)$ when $x$ ranges $[1 081 080, 10^{10} ]$, where $\mathscr{M }(x, 4)$ is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that $|\Delta(x)| \le 0.397 x^{1/2}$ when $x \ge 5 560$ and that $|\Delta(x)| \le 0.764 x^{1/3}\log x$ when $x\ge 9 995$. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning

The Kaplansky condition and rings of almost stable range 1

Abstract: We present some variants of the Kaplansky condition for a K-Hermite ring $R$ to be an elementary divisor ring; for example, a commutative K-Hermite ring $R$ is an EDR iff for any elements $x,y,z\in R$ such that $(x,y)=(1)$, there exists an element $\lambda\in R$ such that $x+\lambda y=uv$, where $(u,z)=(v,1-z)=(1)$. We present an example of a a Bezout domain that is an elementary divisor ring, but it does not have almost stable range 1, thus answering a question of Warren Wm. McGovern.

On the greatest common divisor of a number and its sum of divisors

Abstract: A natural number n is called perfect if σ(n) = 2n and multiply perfect whenever σ(n) is a multiple of n. In 1956, Erdős published improved upper bounds on the counting functions of the perfect and multiply perfect numbers [2]. These estimates were soon superseded by a theorem of Wirsing [15] (Theorem B below), but Erdős’s methods remain of interest as they are applicable to more general questions concerning the distribution of gcd(n, σ(n)). Erdős describes some applications of this type (op. cit.) but omits the proofs. In this paper we prove corrected versions of his results, and we establish some new results in the same direction. For real numbers x ≥ 1 and A ≥ 1, put G(x,A) := #{n ≤ x : gcd(n, σ(n)) > A}.

Asymptotics of complete Kahler metrics of finite volume on quasiprojective manifolds

Abstract: Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later time by showing the associated potential function is smooth up to the boundary on the compactification $\tX$. However, when the divisor $\bD$ is smooth with $K_{\bX}+[\bD]>0$ and the Ricci flow converges to a Kahler-Einstein metric, we show that this Kahler-Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary.

Big arithmetic divisors on the projective spaces over Z

Abstract: In this paper, we observe several properties of an arithmetic divisor D¯ on PZn and give the exact form of the Zariski decomposition of D¯ on PZ1. Further, we show that, if n≥2 and D¯ is big and non-nef, then for any birational morphism f:X→PZn of projective, generically smooth, and normal arithmetic varieties, we cannot expect a suitable Zariski decomposition of f∗(D¯). We also give a concrete construction of Fujita’s approximation of D¯.

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Abstract: Let h be a positive integer and S = {x 1, … , x h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x σ(1) ∣ … ∣ x σ(h) for a permutation σ of {1, … , h}. If the set S can be partitioned as S = S 1 ∪ S 2 ∪ S 3, where S 1, S 2 and S 3 are divisor chains and each element of S i is coprime to each element of S j for all 1 ≤ i < j ≤ 3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x i , x j ) a of the greatest common divisor of x i and x j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S a ). The ath power least common multiple (LCM) matrix [S a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1 ∈ S. We show that if a ∣ b, then the ath power GCD matrix (S a ) (resp., the ath power LCM matrix [S a ]) divides the bth power GCD matrix (S b ) (resp., the bth power LCM matrix [S b ]) ...

A Sharkovsky theorem for vertex maps on trees

Abstract: Let T be a tree with n vertices. Let be continuous and suppose that the n vertices form a periodic orbit under f. We show: If n is not a divisor of 2 k then f has a periodic point with period 2 k . If , where is odd and , then f has a periodic point with period 2 p r for any . The map f also has periodic orbits of any period m where m can be obtained from n by removing ones from the right of the binary expansion of n and changing them to zeroes. Conversely, given any n, there is a tree with n vertices and a map f such that the vertices form a periodic orbit and f has no other periods apart from the ones given above.

Affine SU(N) algebra from wall-crossings

Abstract: We study the relation between the instanton counting on ALE spaces and the BPS state counting on a toric Calabi-Yau three-fold. We put a single D4-brane on a divisor isomorphic to A_{N-1}-ALE space in the Calabi-Yau three-fold, and evaluate the discrete changes of BPS partition function of D4-D2-D0 states in the wall-crossing phenomena. In particular, we find that the character of affine SU(N) algebra naturally arises in wall-crossings of D4-D2-D0 states. Our analysis is completely based on the wall-crossing formula for the d=4, N=2 supersymmetric theory obtained by dimensionally reducing the Calabi-Yau three-fold.

Finite Groups with Four Conjugacy Class Sizes

Abstract: We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.

Residuation of linear series and the effective cone of Cd.

Abstract: We obtain new information about divisors on the dth symmetric power Cd of a general curve C of genus g ≥ 4. This includes a complete description of the effective cone of Cg−1 and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of Cd in the range g+1 ≤ d ≤ g − 2. We also obtain, for each g ≥ 5, a divisor on Cg−1 with non-equidimensional stable base locus. For a general hyperelliptic curve C of genus g, we obtain a complete description of the effective cone of Cd for 2 ≤ d ≤ g and an integral divisor on Cg−1 which has non-integral volume whenever g is not a power of 2. 1. Introduction. Let C be a smooth complex projective algebraic curve, and let d ≥ 1 be an integer. The dth symmetric power Cd := C d /Sd is a smooth d- dimensional complex projective variety which is a fine moduli space parametriz- ing effective divisors of degree d on C. In this paper we study the cone of effective divisors on Cd and its refinements. There are two natural divisor classes on Cd. For each p ∈ C the image of the embedding

On Eckl's pseudo-effective reduction map

Abstract: Suppose that X is a complex projective variety and L is a pseudo-effective divisor. A numerical reduction map is a quotient of X by all subvarieties along which L is numerically trivial. We construct two variants: the L-trivial reduction map and the pseudo-effective reduction map of [Eckl05]. We show that these maps capture interesting geometric properties of L and use them to analyze abundant divisors.

On the Hidden Shifted Power Problem

Abstract: We consider the problem of recovering a hidden element $s$ of a finite field $\F_q$ of $q$ elements from queries to an oracle that for a given $x\in \F_q$ returns $(x+s)^e$ for a given divisor $e\mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm, for example, they use substantially fewer queries to the oracle.

On higher-power moments of the error term for the divisor problem with congruence conditions

Abstract: For a positive integer n, the divisor function with congruence conditions d(n; l 1, M 1, l 2, M 2) denotes the number of factorizations n = n 1 n 2, where each of the factors \({n_i\in\mathbb{N}}\) belongs to a prescribed congruence class l i modulo M i (i = 1, 2). In this paper we study the higher power moments of the error term in the asymptotic formula of \({\sum olimits_{n\leq M_1M_2x}d(n;l_1,M_1,l_2,M_2)}\) .

On the general additive divisor problem

Abstract: We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $\zeta(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.

F-singularities via alterations

Abstract: For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map $\pi_* \omega_Y \to \omega_X$ for every such alteration $\pi \: Y \to X$. Furthermore, building on work of B. Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-Viehweg vanishing theorems in the characteristic $p$ setting without assuming $W2$ lifting, and show that these are strong enough in some applications to extend sections.

Negative curves on surfaces and applications to hermitian locally symmetric spaces

Abstract: Let $S$ be a smooth projective surface over a field $k$. In this paper we show that if $\mathcal{F}$ is a set of irreducible curves on $S$ such that sufficiently many curves in $\mathcal{F}$ have negative self-intersection, then there exist two curves $C_1, C_2 \in \mathcal{F}$ and integers $a, b\ge 1$ such that the effective divisor $a C_1 + b C_2$ has positive self-intersection. When $k = \mathbb{C}$, this implies that the image of the map $\pi_1(C_1 \cup C_2) \to \pi_1(S)$ of topological fundamental groups has finite index and that the Albanese variety of $S$ is a quotient of the direct sum of the Jacobians of the normalizations of $C_1$ and $C_2$. We then give some applications of these results to the geometry of smooth projective surfaces uniformized by either the product of two upper half planes or the complex hyperbolic plane.

An elementary semi-ampleness result for log canonical divisors

Abstract: If the log canonical divisor on a projective variety with only Kawamata log terminal singularities is numerically equivalent to some semi-ample Q-divisor, then it is semi-ample.

Parallel multiple precision division by a single precision divisor

Abstract: We report an algorithm for division of a multi-precision integer by a single-precision value using a graphics processing unit (GPU). Our algorithm combines a parallel version of Jebelean's exact division algorithm with a left-to-right algorithm for computing the borrow chain, to relax the requirement of exactness. We also employ Takahashi's recently reported cyclic reduction technique [10] for GPU division to further enhance performance. The result is that our algorithm is asymptotically faster, at O(n/p + log p), than Takahashi's algorithm at O(n/p log p). We report results for dividends with precisions of 1024, 2048, and 4096 bits running on an NVIDIA GTX 480, and show that, for non-constant divisors, our algorithm is 20% slower at 1024 bits (due to startup overhead), by 2048 we are 40% faster, and at 4096 bits we are able to run 2.5 times faster. For division by constants, with precomputed tables, our algorithm is faster at all sizes with a speedup ranging from 2.3 to 6 times faster.

A New Compact SD2 Positive Integer Triangular Array Division Circuit

Abstract: Division is the highest latency arithmetic operation in present digital architectures and high-performance computing systems; as such drives the demand for efficient hardware division units. Accordingly, this paper proposes a novel architecture for a nonrestoring divisor based on the radix-2 signed-digit (SD2) representation. This notation has been chosen to achieve fast computation, as proposed by Avizienis (IEEE Transactions on Electronic Computers, vol. EC-10, no. 3, pp. 389-400, Sep. 1961), but the architecture presented in this paper, due to its structure and the definition of the cell implementing its architecture, saves area as well. The proposed divisor architecture is able to achieve a delay of order , similar to the solution presented by Takagi (IEICE Transactions on Fundamentals of Electronics, Communications, and Computer Sciences, E89-A, no. 10, pp. 2874-2881, 2006) being considered as the state of the art, instead of other solutions that give growth. This is in line with the fact that even if our carry-chains have a less impact on the circuit the basic cell is larger compared to the one proposed by Takagi Our cells are larger that those proposed in literature, considering them as single circuit, but considering the overall structure there is a saving of some 40% in the number of gates and a gain of 55% in terms of power saving when compared with the state of the art.

Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

Abstract: We present a complete set of efficient explicit formulas for arithmetic in the degree $0$ divisor class group of a genus two real hyperelliptic curve given in affine coordinates. In addition to formulas suitable for curves defined over an arbitrary finite field, we give simplified versions for both the odd and the even characteristic cases. Formulas for baby steps, inverse baby steps, divisor addition, doubling, and special cases such as adding a degenerate divisor are provided, with variations for divisors given in reduced and adapted basis. We describe the improvements and the correctness together with a comprehensive analysis of the number of field operations for each operation. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.