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Showing papers on "Divisor published in 2015"


Journal ArticleDOI
TL;DR: In particular, Amini and Caporaso as mentioned in this paper showed that the rank of a divisor cannot go down under specialization from a complete nonsingular curve to a regular family of semistable curves, which preserves degrees and linear equivalence.
Abstract: A metrized complex of algebraic curves over an algebraically closed field $$\kappa $$ is, roughly speaking, a finite metric graph $$\Gamma $$ together with a collection of marked complete nonsingular algebraic curves $$C_v$$ over $$\kappa $$ , one for each vertex $$v$$ of $$\Gamma $$ ; the marked points on $$C_v$$ are in bijection with the edges of $$\Gamma $$ incident to $$v$$ . We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over $$\kappa $$ with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve $$X$$ defined over a non-Archimedean field $$\mathbb {K}$$ , together with a strongly semistable model $$\mathfrak {X}$$ for $$X$$ over the valuation ring $$R$$ of $$\mathbb {K}$$ , we define a corresponding metrized complex $$\mathfrak {C}\mathfrak {X}$$ of curves over the residue field $$\kappa $$ of $$\mathbb {K}$$ and a canonical specialization map $$\tau ^{\mathfrak {C}\mathfrak {X}}_*$$ from divisors on $$X$$ to divisors on $$\mathfrak {C}\mathfrak {X}$$ which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from $$X$$ to $$\mathfrak {C}\mathfrak {X}$$ . As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a $$\mathfrak {g}^r_d$$ in a regular family of semistable curves is a limit $$\mathfrak {g}^r_d$$ on the special fiber.

85 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the skeleton of the Jacobian of the Berkovich analytification is canonically isomorphic to the Jacobians of the skeleton as principally polarized tropical abelian varieties, and that the tropicalization of a classical Abel-Jacobi map is a tropical Abel Jacobian map.
Abstract: Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.

76 citations


Journal ArticleDOI
TL;DR: The existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on a Ricci-flat Kahler manifold with one end converges at an exponential rate to a compact Ricciflat manifold with a single end was shown in this paper.
Abstract: Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure theorems for $M$; in particular we show that there is no loss of generality in assuming that $M$ is simply-connected and irreducible with $\mathrm{Hol}(M) = \mathrm{SU}(n)$, where $n$ is the complex dimension of $M$. If $n \gt 2$ we then show that there exists a projective orbifold $\overline{M}$ and a divisor $\overline{D} \in \lvert -K_{\overline{M}} \rvert$ with torsion normal bundle such that $\overline{M}$ is biholomorphic to $\overline{M} \setminus \overline{D}$, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting.We give examples where $\overline{M}$ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair $\overline{M} \setminus \overline{D}$ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi–Yau metrics on $\overline{M} \setminus \overline{D}$.

60 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the three-dimensional effective action obtained by reducing eleven-dimensional supergravity with higher-derivative terms on a background solution including a warp-factor, an eight-dimensional compact manifold, and fluxes.
Abstract: We study the three-dimensional effective action obtained by reducing eleven-dimensional supergravity with higher-derivative terms on a background solution including a warp-factor, an eight-dimensional compact manifold, and fluxes. The dynamical fields are Kahler deformations and vectors from the M-theory three-form. We show that the potential is only induced by fluxes and the naive contributions obtained from higher-curvature terms on a Calabi-Yau background vanish once the back-reaction to the full solution is taken into account. For the resulting three-dimensional action we analyse the Kahler potential and complex coordinates and show compatibility with $$ \mathcal{N}=2 $$ supersymmetry. We argue that the higher-order result is also compatible with a no-scale condition. We find that the complex coordinates should be formulated as divisor integrals for which a non-trivial interplay between the warp-factor terms and the higher-curvature terms allow a derivation of the moduli space metric. This leads us to discuss higher-derivative corrections to the M5-brane action.

56 citations


Journal ArticleDOI
TL;DR: The notion of normal crossing divisors from algebraic geometry has been introduced in this paper, where the authors define the GW invariant of a symplectic manifold X relative to such a divisor V. The invariants are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16].

43 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that any divisor on the chain of loops that is rational over the value group lifts to a divisors of the same rank on, confirming a conjecture of Cools, Draisma, Robeva, and the third author.
Abstract: Let be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on , confirming a conjecture of Cools, Draisma, Robeva, and the third author.

40 citations


Book ChapterDOI
29 Nov 2015
TL;DR: In this paper, the problem of finding small solutions to a collection of linear equations modulo an unknown divisor p for a known composite integer N was revisited, and several generalizations of the above equations were proposed.
Abstract: We revisit the problem of finding small solutions to a collection of linear equations modulo an unknown divisor p for a known composite integer N. In CaLC 2001, Howgrave-Graham introduced an efficient algorithm for solving univariate linear equations; since then, two forms of multivariate generalizations have been considered in the context of cryptanalysis: modular multivariate linear equations by Herrmann and May Asiacrypt'08 and simultaneous modular univariate linear equations by Cohn and Heninger ANTS'12. Their algorithms have many important applications in cryptanalysis, such as factoring with known bits problem, fault attacks on RSA signatures, analysis of approximate GCD problem, etc. In this paper, by introducing multiple parameters, we propose several generalizations of the above equations. The motivation behind these extensions is that some attacks on RSA variants can be reduced to solving these generalized equations, and previous algorithms do not apply. We present new approaches to solve them, and compared with previous methods, our new algorithms are more flexible and especially suitable for some cases. Applying our algorithms, we obtain the best analytical/experimental results for some attacks on RSA and its variants, specifically,We improve May's results PKC'04 on small secret exponent attack on RSA variant with moduli $$N = p^rq$$$$r\ge 2$$.We experimentally improve Boneh et al.'s algorithm Crypto'98 on factoring $$N=p^rq$$$$r\ge 2$$ with known bits problem.We significantly improve Jochemsz-May' attack Asiacrypt'06 on Common Prime RSA.We extend Nitaj's result Africacrypt'12 on weak encryption exponents of RSA and CRT-RSA.

38 citations


Journal ArticleDOI
TL;DR: Conlon and Hein this article showed that if D is a suborbifold divisor in X, then, applying results from their previous paper, each Kahler class on X contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r petertodd −1-e fixme ) if X is smooth.
Abstract: Let X be a compact Kahler orbifold without $${\mathbb{C}}$$ -codimension-1 singularities. Let D be a suborbifold divisor in X such that $${D \supset {\rm Sing}(X)}$$ and −pK X = q[D] for some $${p, q \in \mathbb{N}}$$ with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler–Einstein, then, applying results from our previous paper (Conlon and Hein, Duke Math J, 162:2855–2902, 2013), we show that each Kahler class on $${X \setminus D}$$ contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r −1-e ) if X is smooth. This provides a definitive version of a theorem of Tian and Yau (Invent Math, 106:27–60, 1991). (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128)) when $${X = {\rm Bl}_p \mathbb{P}^{3}}$$ and $${D = {\rm Bl}_{p_1,p_2} \mathbb{P}^{2}}$$ is the strict transform of a smooth quadric through p in $${\mathbb{P}^3}$$ . Here D is no longer Kahler–Einstein, but the normal $${\mathbb{S}^1}$$ -bundle to D in X admits an irregular Sasaki–Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi–Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

34 citations


Journal ArticleDOI
TL;DR: A general construction of classes of linear codes from o-polynomials and their weight distribution proving that all of them are constant weight codes is presented, and it is shown that the hyperovals of PG2(2m) from finite projective geometry provide new minimal codes.
Abstract: The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and $$q$$q-ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. We show that the hyperovals of $$PG_{2}(2^m)$$PG2(2m) from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of $$2^r$$2r-ary ($$r$$r being a divisor of $$m$$m) simplex linear codes (whose duals are the perfect $$2^r$$2r-ary Hamming codes) over an extension field $${\mathbb F}_{2^{r}}$$F2r of $${\mathbb F}_{2^{}}$$F2. The following diagram gives an indication of the main topics and interconnections arising in this paper. [Figure not available: see fulltext.]

33 citations


Posted Content
TL;DR: In this article, it was shown that a fiber space with non-normal fibers is uniruled and that general fibers of Mori fiber spaces are rationally chain connected, and a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field was obtained.
Abstract: Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{{\rm red}}$. In this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.

33 citations


Journal ArticleDOI
TL;DR: In this article, the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements are studied.
Abstract: We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

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TL;DR: In this article, the authors established convolution sums of functions for the divisor sums for certain values of the Glaisher constant, such as the sum of the number of representations of a function as a sum of triangular numbers.
Abstract: One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

Journal ArticleDOI
TL;DR: Goldman and Bowden as mentioned in this paper showed that every geometric representation of a closed surface group into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class.
Abstract: Let $$\Gamma _g$$ denote the fundamental group of a closed surface of genus $$g \ge 2$$ . We show that every geometric representation of $$\Gamma _g$$ into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of $$\Gamma _g$$ into $${{\mathrm{Homeo}}}_+(S^1)$$ . Precisely, for each nontrivial divisor $$k$$ of $$2g-2$$ , there are at least $$|k|^{2g} + 1$$ components containing representations with Euler number $$\frac{2g-2}{k}$$ . Our methods apply to representations of surface groups into finite covers of $${{\mathrm{PSL}}}(2,\mathbb {R})$$ and into $${{\mathrm{Diff}}}_+(S^1)$$ as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.

Journal ArticleDOI
TL;DR: For any Legendrian link, in (R^3, \ker(dz-y\,dx)) the invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L.
Abstract: For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the case q =2, we show the augmentation numbers, Aug_m(L,q), are determined by specializing the m-graded ruling polynomial, R^m_L(z), at z = q^{1/2}-q^{-1/2}. As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.

Posted Content
TL;DR: In this paper, it was shown that for some constants $c, $c_1,$c_2,$ and $c-3, the triple divisor function can be approximated by the sum of the number of solutions of the equation d_1d_2d_3=n$ in natural numbers.
Abstract: Let $\tau_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.

Journal ArticleDOI
TL;DR: In this article, the smallest known complete arcs in affine spaces of dimension n ≥ 4 were constructed from singular cubic curves with a characteristic greater than three points, where n ≥ 1 4 q 1/4.
Abstract: Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$ F q with characteristic greater than three are constructed from singular cubic curves. For $$m$$ m a divisor of $$q+1$$ q + 1 or $$q-1$$ q - 1 , complete plane arcs of size approximately $$q/m$$ q / m are obtained, provided that $$(m,6)=1$$ ( m , 6 ) = 1 and $$m<\frac{1}{4}q^{1/4}$$ m < 1 4 q 1 / 4 . If in addition $$m=m_1m_2$$ m = m 1 m 2 with $$(m_1,m_2)=1$$ ( m 1 , m 2 ) = 1 , then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$ m 1 + m 2 m q N / 2 points are described. These results substantially widen the spectrum of $$q$$ q s for which complete arcs in $$AG(2,q)$$ A G ( 2 , q ) of size approximately $$q^{3/4}$$ q 3 / 4 can be constructed. Complete caps in $$AG(N,q)$$ A G ( N , q ) with roughly $$q^{(4N-1)/8}$$ q ( 4 N - 1 ) / 8 points are also provided. For infinitely many $$q$$ q s, these caps are the smallest known complete caps in $$AG(N,q)$$ A G ( N , q ) , $$N \equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) .

Posted Content
TL;DR: In this article, it was shown that for a semi-ample divisor, there exists an effective log-concatenative projective log-canonical pair over finite fields.
Abstract: Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $\Delta' \sim_{\mathbb Q} \Delta+D$ such that $(X, \Delta')$ is log canonical if there exists a log resolution of $(X, \Delta)$.

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TL;DR: In this paper, it was shown that Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ implies the existence of a divisor that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N.
Abstract: We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log discrepancy of (X,J) at x and such that its discrepancy k_E is bounded above by N. We show that this implies Shokurov's ACC conjecture for minimal log discrepancies on a fixed klt germ and give some partial results towards the conjecture.

Journal ArticleDOI
TL;DR: In this article, the moduli space of complete quadric surfaces has been studied using tools from the minimal model program (MMP) and it has been shown that it is a Mori dream space.
Abstract: Let $X$ be the moduli space of complete $(n-1)$-quadrics. In this thesis, we study the birational geometry of $X$ using tools from the minimal model program (MMP). In Chapter $1$, we recall the definition of the space $X$ and summarize our main results in Theorems A, B and C. \medskip In Chapter $2$, we examine the codimension-one cycles of the space $X$, and exhibit generators for Eff$(X)$ and Nef$(X)$ (Theorem A), the cone of effective divisors and the cone of nef divisors, respectively. This result, in particular, allows us to conclude the space $X$ is a Mori dream space. \medskip In Chapter $3$, we study the following question: when does a model of $X$, defined as $X(D):= \mathrm{Proj}(\bigoplus_{m\ge 0}H^0(X,mD))$, have a moduli interpretation? We describe such an interpretation for the models $X(H_k)$ (Theorem B), where $H_k$ is any generator of the nef cone $\mathrm{Nef}(X)$. In the case of complete quadric surfaces there are 11 birational models $X(D)$ (Theorem B), where $D$ is a divisor in the movable cone $\mathrm{Mov}(X)$, and among which we find a moduli interpretation for seven of them. \medskip Chapter 4 outlines the relation of this work with that of Semple \cite{SEM}, \cite{SEMII} as well as future directions of research.

Journal ArticleDOI
TL;DR: The Schutzenberger category of a semigroup was introduced in this paper, which stands in relation to the Karoubi envelope and the Cauchy completion of the semigroup.
Abstract: In this paper we introduce the Schutzenberger category \({\mathbb {D}}(S)\) of a semigroup \(S\). It stands in relation to the Karoubi envelope (or Cauchy completion) of \(S\) in the same way that Schutzenberger groups do to maximal subgroups and that the local divisors of Diekert do to the local monoids \(eSe\) of \(S\) with \(e\in E(S)\). In particular, the objects of \({\mathbb {D}}(S)\) are the elements of \(S\), two objects of \({\mathbb {D}}(S)\) are isomorphic if and only if the corresponding semigroup elements are \({\fancyscript{D}}\)-equivalent, the endomorphism monoid at \(s\) is the local divisor in the sense of Diekert and the automorphism group at \(s\) is the Schutzenberger group of the \({\fancyscript{H}}\)-class of \(s\) in \(S\). This makes transparent many well-known properties of Green’s relations. The paper also establishes a number of technical results about the Karoubi envelope and Schutzenberger category that were used by the authors in a companion paper on syntactic invariants of flow equivalence of symbolic dynamical systems.

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TL;DR: In this article, it was shown that for a large class of stacks one typically encounters, this description does indeed characterize them and that each such stack can be described in terms of two simple procedures applied iteratively to its coarse space.
Abstract: In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify the stack. Our main result shows that for a large class of stacks one typically encounters, this description does indeed characterize them. Moreover, we prove that each such stack can be described in terms of two simple procedures applied iteratively to its coarse space: canonical stack constructions and root stack constructions. More precisely, if $\mathcal X$ is a smooth separated tame Deligne-Mumford stack of finite type over a field $k$ with trivial generic stabilizer, it is completely determined by its coarse space $X$ and the ramification divisor (on $X$) of the coarse space morphism $\pi\colon \mathcal X \to X$. Therefore, to specify such a stack, it is enough to specify a variety and the orders of the stabilizers of codimension 1 points. The group structures, as well as the stabilizer groups of higher codimension points, are then determined.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a divisor has normal crossings if and only if it is a free divisors, has a radical Jacobian ideal and a smooth normalization.
Abstract: The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a divisor (=hypersurface) has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito’s theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found. Finally, we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.

Journal ArticleDOI
TL;DR: Li, Helleseth, Tang and Tang as mentioned in this paperinite field polynomials of the form (x,b,c,u,v \in \mathbb{F}_{q^2}), where $d = 2, 3, 4, 6, 6.
Abstract: Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\mathbb{F}_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{\frac{q^2 -1}{d}+1} -bx$ over $\mathbb{F}_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form \[ f(x)=(ax^{q} +bx +c)^r \phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~\text{over $\mathbb{F}_{q^2}$}, \] where $a,b,c,u,v \in \mathbb{F}_{q^2}$, $r \in \mathbb{Z}^{+}$, $\phi(x)\in \mathbb{F}_{q^2}[x]$ and $d$ is an arbitrary positive divisor of $q^2-1$. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary--Ghioca--Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether $f(x)$ permutes $\mathbb{F}_{q^2}$ to that of verifying whether two more polynomials permute two subsets of $\mathbb{F}_{q^2}$. As a consequence, we find a series of simple conditions for $f(x)$ to be a PP of $\mathbb{F}_{q^2}$. These results unify and generalize some known classes of PPs.

Journal ArticleDOI
TL;DR: In this article, an infinite family of irreducible homogeneous free divisors in K[x, y, z] is constructed, and the set of monomials X such that the general polynomial supported on X is a non-free divisor is identified.
Abstract: An infinite family of irreducible homogeneous free divisors in K[x, y, z] is constructed. Indeed, we identify sets of monomials X such that the general polynomial supported on X is a free divisor.

Journal ArticleDOI
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 Bykovskiĭ and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindelof hypothesis is assumed.

Journal ArticleDOI
TL;DR: The average value of the divisor function is known to be evenly distributed over arithmetic progressions for all polynomials that are a little smaller than 2/3.
Abstract: We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes $p$ and any fixed integer $k\ge 7$.

Journal ArticleDOI
Yaim Cooper1
TL;DR: In this paper, the coarse moduli schemes associated to the smooth proper Deligne-Mumford stacks were studied and shown to be projective, unirational, and have Picard number 2.
Abstract: Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When \(n \ge 2\), the space \(\overline{Q}_{g}({\mathbb {P}}^{n-1},d) = \overline{Q}_{g}(G(1,n),d)\) compactifies the space of degree \(d\) maps of smooth genus \(g\) curves to \({\mathbb {P}}^{n-1}\), while \(\overline{Q}_{g}(G(1,1),d) \simeq \overline{M}_{1, d \cdot \epsilon }/S_d\) is a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne–Mumford stacks \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\), for all \(n \ge 1\). We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, the cones of ample divisors, and in the case \(n=1\) the cones of effective divisors. We conclude that \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\) is Fano if and only if \(n(d-1)(d+2) < 20\). Moreover, we show that \({\overline{Q}}_{1}({\mathbb {P}}^{n-1},d)\) is a Mori Fiber space for all \(n,d\), hence always minimal in the sense of the minimal model program. In the case \(n=1\), we write in addition a closed formula for the Poincare polynomial.

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TL;DR: In this paper, it was shown that the index of an Eisenstein maximal ideal contained in a cuspidal divisor annihilated by the cusidal subgroup of the Jacobian $J_0(N) for a square-free integer $N>6 was 0.
Abstract: Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a square-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}] eq 0$. To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$ contained in $\mathfrak{m}$ by computing the order of a cuspidal divisor annihilated by $\mathcal{I}$.

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TL;DR: In this paper, the authors considered the moduli stack of arithmetic genus G(n-g,n) with smooth marked points and nonzero tangent vectors, such that the divisor $p_1+\ldots+p_n$ is nonspecial (has no $h^1$) and ample.
Abstract: In this paper for each $n\ge g\ge 0$ we consider the moduli stack $\widetilde{\mathcal U}^{ns}_{g,n}$ of curves $(C,p_1,\ldots,p_n,v_1,\ldots,v_n)$ of arithmetic genus $g$ with $n$ smooth marked points $p_i$ and nonzero tangent vectors $v_i$ at them, such that the divisor $p_1+\ldots+p_n$ is nonspecial (has no $h^1$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{\mathcal U}^{ns}_{g,n}$ with a certain relative moduli of $A_\infty$-structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$.

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TL;DR: In this paper, it was shown that the restricted tangent bundle of a general curve of genus g, equipped with a general degree d map f to P^r, satisfies the property of interpolation.
Abstract: Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f : C \to P^r so that f(p_i) = q_i for all i. This is a consequence of our main theorem, which states that the restricted tangent bundle f^* T_{P^r} of a general curve of genus g, equipped with a general degree d map f to P^r, satisfies the property of interpolation (i.e.\ that for a general effective divisor D of any degree on C, either H^0(f^* T_{P^r}(-D)) = 0 or H^1(f^* T_{P^r}(-D)) = 0). We also prove an analogous theorem for the twist f^* T_{P^r}(-1).