Showing papers on "Divisor published in 2017"

Double ramification cycles on the moduli spaces of curves

Abstract: Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $ u$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu, u)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu, u)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu, u)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu= u=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

New MDS Self-Dual Codes From Generalized Reed—Solomon Codes

Abstract: Both Maximum Distance Separable and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining the existence of $q$ -ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where $q$ is even. This paper focuses on the case where $q$ is odd. We construct a few classes of new MDS self-dual codes through generalized Reed–Solomon codes. More precisely, we show that for any given even length $n$ , we have a $q$ -ary MDS code as long as $q\equiv 1\bmod {4}$ and $q$ is sufficiently large (say $q\ge 4^{n}\times n^{2})$ . Furthermore, we prove that there exists a $q$ -ary MDS self-dual code of length $n$ if $q=r^{2}$ and $n$ satisfies one of the three conditions: 1) $n\le r$ and $n$ is even; 2) $q$ is odd and $n-1$ is an odd divisor of $q-1$ ; and 3) $r\equiv 3\mod {4}$ and $n=2tr$ for any $t\le (r-1)/2$ .

TruncApp: A truncation-based approximate divider for energy efficient DSP applications

Abstract: In this paper, we present a high speed yet energy efficient approximate divider where the division operation is performed by multiplying the dividend by the inverse of the divisor. In this structure, truncated value of the dividend is multiplied exactly (approximately) by the approximate inverse value of divisor. To assess the efficacy of the proposed divider, its design parameters are extracted and compared to those of a number of prior art dividers in a 45nm CMOS technology. Results reveal that this structure provides 66% and 52% improvements in the area and energy consumption, respectively, compared to the most advanced prior art approximate divider. In addition, delay and energy consumption of the division operation are reduced about 94.4% and 99.93%, respectively, compared to those of an exact SRT radix-4 divider. Finally, the efficacy of the proposed divider in image processing application is studied.

Singularities of General Fibers and the LMMP

Abstract: We use the theory of foliations to study the relative canonical divisor of a normalized inseparable base-change. Our main technical theorem states that it is linearly equivalent to a divisor with positive integer coefficients divisible by $p-1$. We deduce many consequences about the fibrations of the minimal model program: for example the general fibers of terminal $3$-fold Mori fiber spaces are normal in characteristic $p\geq 5$ and smooth in characteristic $p\geq 11$.

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Abstract: We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed.

Derived categories of cyclic covers and their branch divisors

Abstract: Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover \(X \rightarrow Y\) ramified over a divisor \(Z \subset Y\). We construct semiorthogonal decompositions of \(\mathrm {D^b}(X)\) and \(\mathrm {D^b}(Z)\) with distinguished components \({\mathcal {A}}_X\) and \({\mathcal {A}}_Z\) and prove the equivariant category of \({\mathcal {A}}_X\) (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into \(n-1\) copies of \({\mathcal {A}}_Z\). As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

The Hodge Numbers of Divisors of Calabi-Yau Threefold Hypersurfaces

Abstract: We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$. We give a formula for $h^i({\cal{O}}_D)$ in terms of combinatorial data. Moreover, we construct a CW complex $\mathscr{P}_D$ such that $h^i({\cal{O}}_D)=h_i(\mathscr{P}_D)$. We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large $h^{1,1}$. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with $h^{1,1}=491$. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi-Yau hypersurfaces.

Newton–Okounkov bodies sprouting on the valuative tree

Abstract: Given a smooth projective algebraic surface X, a point $$O\in X$$ and a big divisor D on X, we consider the set of all Newton–Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E, p) which is infinitely near O, in the sense that there is a sequence of blowups $$X' \rightarrow X$$ , mapping the smooth, irreducible rational curve $$E\subset X'$$ to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton–Okounkov bodies as (E, p) varies, focusing on the case $$X=\mathbb {P}^2$$ .

The Geometry of F$_4$-Models

Abstract: We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometries F$_4$-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram $\widetilde{\text{F}}_4^t$. These geometries are used in string theory to model gauge theories with the exceptional Lie group F$_4$ on a smooth divisor $S$ of the base. Starting with a singular Weierstrass model of an F$_4$-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over $S$. These are $\mathbb{P}^1$-bundles over $S$ or double covers of $\mathbb{P}^1$-bundles over $S$. We compute basic topological invariants such as the double and triple intersection numbers of the fibral divisors and the Euler characteristic of the F$_4$-model. In the case of Calabi-Yau threefolds, we compute the linear form induced by the second Chern class and the Hodge numbers. We also explore the meaning of these geometries for the physics of gauge theories in five and six-dimensional minimal supergravity theories with eight supercharges. We also introduce the notion of "frozen representations" and explore the role of the Stein factorization in the study of fibral divisors of elliptic fibrations.

Factorization for stacks and boundary complexes

Abstract: We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $\Delta(X,D)$ depends only on $U$.

On the regularity problem of complex Monge–Ampere equations with conical singularities

Abstract: In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2\pi)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kahler-Einstein equations actually possess maximum regularity, which means the metrics are actually Holder continuous in the singular polar coordinates. This shows the weak Kahler-Einstein metrics constructed by Guenancia-Paun \cite{GP}, and independently by Yao \cite{GT}, are all actually strong-conical Kahler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kahler-Ricci flat metrics defined over $\C^{n}$, which depends on a Calderon-Zygmund theory in the conical setting.

Realizability of tropical canonical divisors

Abstract: We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and a divisor $D$ in the canonical linear system on $\Gamma$, we give a purely combinatorial condition to decide whether there is a smooth curve $X$ over a non-Archimedean field whose stable reduction has $\Gamma$ as its dual tropical curve together with a effective canonical divisor $K_X$ that specializes to $D$. Along the way, we develop a moduli-theoretic framework to understand Baker's specialization of divisors from algebraic to tropical curves as a natural toroidal tropicalization map in the sense of Abramovich-Caporaso-Payne.

Arithmetic ampleness and an arithmetic Bertini theorem

Abstract: Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal L}^{\otimes n}$ that define an irreducible divisor on $\mathcal X$ tends to $1$ as $n$ tends to $\infty$. We prove variants of this statement for schemes mapping to such an $\mathcal X$. On the way to these results, we discuss some general properties of arithmetic ampleness, including restriction theorems, and upper bounds for the number of effective sections of hermitian line bundles on arithmetic varieties.

New cubic fourfolds with odd-degree unirational parametrizations

Abstract: We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor $\mathcal{C}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that $\mathcal{C}_{42}$ is uniruled.

On relative and overconvergent de Rham–Witt cohomology for log schemes

Abstract: We construct the relative log de Rham–Witt complex. This is a generalization of the relative de Rham–Witt complex of Langer–Zink to log schemes. We prove the comparison theorem between the hypercohomology of the log de Rham–Witt complex and the relative log crystalline cohomology in certain cases. We construct the p-adic weight spectral sequence for relative proper strict semistable log schemes. When the base log scheme is a log point, We show it degenerates at \(E_2\) after tensoring with the fraction field of the Witt ring. We also extend the definition of the overconvergent de Rham–Witt complex of Davis–Langer–Zink to log schemes (X, D) associated with smooth schemes with simple normal crossing divisor over a perfect field. Finally, we compare its hypercohomology with the rigid cohomology of \(X{\setminus }D\).

The maximum number of lines lying on a K3 quartic surface

Abstract: We show that there cannot be more than 64 lines on a quartic surface with isolated rational double points over an algebraically closed field of characteristic $$p e 2,\,3$$ , thus extending Segre–Rams–Schutt theorem. Our proof offers a deeper insight into the triangle-free case and takes advantage of a special configuration of lines, thereby avoiding the technique of the flecnodal divisor. We provide several examples of non-smooth K3 quartic surfaces with many lines.

Moments of averages of generalized Ramanujan sums

Abstract: Let $$\beta $$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by $$\begin{aligned} c_{q,\beta }(n) := \sum \limits _{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \end{aligned}$$ where h ranges over the non-negative integers less than $$q^{\beta }$$ such that h and $$q^{\beta }$$ have no common $$\beta $$ -th power divisors other than 1. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of $$c_{q,\beta }(n)$$ by computing the k-th moments of the average value of $$c_{q,\beta }(n)$$ . In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan’s formula.

Correspondences between convex geometry and complex geometry

Abstract: We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.

Rankin–Selberg local factors modulo \ell

Abstract: After extending the theory of Rankin–Selberg local factors to pairs of \(\ell \)-modular representations of Whittaker type, of general linear groups over a non-Archimedean local field, we study the reduction modulo \(\ell \) of \(\ell \)-adic local factors and their relation to these \(\ell \)-modular local factors. While the \(\ell \)-modular local \(\gamma \)-factor we associate with such a pair turns out to always coincide with the reduction modulo \(\ell \) of the \(\ell \)-adic \(\gamma \)-factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour, always dividing the reduction modulo-\(\ell \) of the \(\ell \)-adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. We completely describe \(\ell \)-modular L-factors in the generic case and obtain two simple-to-state nice formulae: Let \(\pi ,\pi '\) be generic \(\ell \)-modular representations; then, writing \(\pi _b,\pi '_b\) for their banal parts, we have $$\begin{aligned} L(X,\pi ,\pi ')=L(X,\pi _b,\pi _b'). \end{aligned}$$ Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that $$\begin{aligned} L(X,\pi ,\pi ')=\mathop {\mathbf {GCD}}(r_{\ell }(L(X,\tau ,\tau '))), \end{aligned}$$ where the divisor is over all integral generic \(\ell \)-adic representations \(\tau \) and \(\tau '\) which contain \(\pi \) and \(\pi '\), respectively, as subquotients after reduction modulo \(\ell \).

Geometry and arithmetic of certain log K3 surfaces

Abstract: Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\mathbb{Z}$ of $X$, the set of integral points $\mathcal{X}(\mathbb{Z})$ is not Zariski dense. We also show that the Brauer-Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is "split".

Connections on Parahoric Torsors over Curves

Abstract: We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--torsor is given by a homomorphism from $\pi_1(X\setminus D)$ to a maximal compact subgroup of $G$, where $D\, \subset\, X$ is the parabolic divisor, if and only if the torsor is polystable.

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

Abstract: We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n \Lambda(n) \Lambda(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying Holder's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates.

A few questions about curves on surfaces

Abstract: In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and \(D\) a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on \(D\) or on \(X\).

KP theory, plabic networks in the disk and rational degenerations of M--curves

Abstract: We complete the program of Ref. [3,5] of connecting totally non-negative Grassmannians to the reality problem in KP finite-gap theory via the assignment of real regular divisors on rational degenerations of M-curves for the class of real regular multi-line soliton solutions of KP II equation whose asymptotic behavior has been combinatorially characterized by Chakravarthy, Kodama and Williams. We use the plabic networks in the disk introduced by Postnikov to parametrize positroid cells in totally nonnegative Grassmannians. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem, and the bicolored graph is the dual of a reducible curve G which is the rational degeneration of a regular M-curve whose genus g equals the number of faces of the network diminished by one. We assign systems of edge vectors to the planar bicolored networks. The system of relations satisfied by them has maximal rank and may be reformulated in the form of edge signatures.. We prove that the components of the edge vectors are rational in the edge weights with subtraction free denominators and provide their explicit expressions in terms of conservative and edge flows. The edge vectors rule the value of the KP wave function at the double points of G, whereas the signatures at the vertices rule the position of the divisor in the ovals. We prove that the divisor satisfies the conditions of Dubrovin-Natanzon for real finite-gap solutions: there is exactly one divisor point in each oval except for the one containing the essential singularity of the wave function. The divisor may be explicitly computed using the linear relations at the vertices of the network. We explain the role of moves and reductions in the transformation of both the curve and the divisor for given soliton data, and apply our construction to some examples.

S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points

Abstract: Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integer coefficients and with at least two roots in the complex numbers, then for every integer $x$ at which $f(x)$ is non-zero, we have (*) $[f(x)]_S\leq c\cdot |f(x)|^d$, where $c$ and $d$ are effectively computable and $d 1/n$, provided we do not require effectivity of $c$. Further, we show that such an inequality does not hold anymore with $d=1/n$ and sufficiently small $c$. In addition we prove a density result, giving for every $\epsilon>0$ an asymptotic estimate with the right order of magnitude for the number of integers $x$ with absolute value at most $B$ such that $f(x)$ has $S$-part at least $|f(x)|^{\epsilon}$. The result of Gross and Vincent, as well as the other results mentioned above, are generalized to values of binary forms and decomposable forms at integral points. Our main tools are Baker type estimates for linear forms in complex and $p$-adic logarithms, the $p$-adic Subspace Theorem of Schmidt and Schlickewei, and a recent general lattice point counting result of Barroero and Widmer.

Equations of hyperelliptic Shimura curves

Abstract: By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves . As a byproduct, we also address the problem of whether a modular form on Shimura curves with a divisor supported on CM divisors can be realized as a Borcherds form, where denotes the quotient of by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

Divisors computing the minimal log discrepancy on a smooth surface

Abstract: We study a divisor computing the minimal log discrepancy on a smooth surface. Such a divisor is obtained by a weighted blow-up. There exists an example of a pair such that any divisor computing the minimal log discrepancy computes no log canonical thresholds.