Showing papers on "Divisor published in 2013"
TL;DR: This work confirms a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to theRank of D on the corresponding metric graph, and constructs an algorithm for computing therank of a Divisor on a tropical curve.
81 citations
TL;DR: In this paper, it was shown that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that vp(ϕ n (α)) > 0a ndvp( ϕ m (α)f 0 for all positive integers m 0 with the stronger condition vp n (β) = 1.
Abstract: Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d> 1, and let α ∈ K be a wandering point such that ϕ n (α) �= 0 for all n> 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that vp(ϕ n (α)) > 0a ndvp(ϕ m (α)) 0 for all positive integers m 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial. Let K be a number field or function field, let ϕ(x) ∈ K(x) be a rational function of degree d> 1, and let α ∈ K. We denote the n-iterate of ϕ as ϕ n . It is often the case that for all but finitely many n, there is a prime that is a divisor of ϕ n (α), but is not a divisor of ϕ m (α )f or any m
59 citations
TL;DR: In this paper, a projective irreducible holomorphic manifold X is considered and the second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing.
Abstract: Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville–Bogomolov pairing. A divisor E on X is called a prime exceptional divisor if E is reduced and irreducible and of negative Beauville–Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology H∗(X,Z), which acts on the second cohomology lattice as the reflection by the cohomology class [E] of E. We then specialize to the case where X is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface, n≥2. We determine the set of classes of exceptional divisors on X. This leads to a determination of the closure of the movable cone of X.
50 citations
Posted Content•
TL;DR: In this article, 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.
49 citations
TL;DR: In this article, it was shown that T (X1;S1;n;m) is given by a formula reminiscent of a Lefschetz xed point formula: the function m 7!
Abstract: Let X1 be a curve of genus g, projective and smooth over Fq. Let S1 X1 be a reduced divisor consisting of N1 closed points of X1. Let (X;S) be obtained from (X1;S1) by extension of scalars to an algebraic closure F of Fq. Fix a primel not dividingq. The pullback by the Frobenius endomorphism Fr ofX induces a permutation Fr of the set of isomorphism classes of rank n irreducible Ql-local systems on X S. It maps to itself the subset of those classes for which the local monodromy at each s2 S is unipotent, with a single Jordan block. Let T (X1;S1;n;m) be the number of xed points of Fr m acting on this subset. Under the assumption that N1 2, we show that T (X1;S1;n;m) is given by a formula reminiscent of a Lefschetz xed point formula: the function m 7! T (X1;S1;n;m) is of the form P ni m for suitable integers ni and \eigenvalues" i. We use Laorgue
46 citations
TL;DR: In this paper, it was shown that the Chow quotient parametrizing configurations of n points in a rational normal curve is isomorphic to the conformal blocks line bundle.
Abstract: We prove that the Chow quotient parametrizing configurations of n points in $\mathbb{P}^d$ which generically lie on a rational normal curve is isomorphic to $\overline{M}_{0,n}$, generalizing the well-known $d = 1$ result of Kapranov. In particular, $\overline{M}_{0,n}$ admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of $\overline{M}_{0,n}$ as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, $\overline{M}_{0,2m}$ is fixed pointwise by the Gale transform when $d=m-1$ so stable curves correspond to self-associated configurations.
42 citations
TL;DR: In this article, a conjectural description for the cone of effective divisors of the Grothendieck-Knudsen moduli space M 0;n of stable rational curves with n marked points is given.
Abstract: We give a conjectural description for the cone of effective divisors of the Grothendieck-Knudsen moduli space M 0;n of stable ra- tional curves with n marked points. Namely, we introduce new combi- natorial structures called hypertrees and show that they give exceptional divisors onM 0;n with many remarkable properties. x1. INTRODUCTION A major open problem inspired by the pioneering work of Harris and Mumford (HM) on the Kodaira dimension of the moduli space of stable curves, is to understand geometry of its birational models, and in particular to describe its cone of effective divisors and a decomposition of this cone into Mori chambers (HK) encoding ample divisors on birational models. Here we study the genus zero case. The moduli spaces M 0;n parame- trize stable rational curves, i.e., nodal trees of P 1 's with n marked points and without automorphisms. For any subsetI of marked points, M 0;n has a natural boundary divisor I whose general element parametrizes stable rational curves with two irreducible components, one marked by points in I and another marked by points inI c . We will introduce new combinatorial objects called hypertrees with an eye towards the following 1.1. CONJECTURE. The effective cone of M 0;n is generated by boundary divisors and by divisorsD (defined below) parametrized by irreducible hypertrees.
37 citations
TL;DR: A generalization of Kawamata's result in arbitrary dimension was shown in this paper. But this result is restricted to Fano manifold of dimension n and index n − 3.
Abstract: Let X be a Fano manifold of dimension n and index n − 3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n = 4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension.
34 citations
Posted Content•
TL;DR: In this paper, the augmented base locus coincides with the exceptional locus (i.e., null locus) for any Cartier divisor on any scheme projective over a field (of any characteristic).
Abstract: We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the augmented base locus generalizing a result of Keel. We also study nef divisors with positive top intersection number, and discuss some problems related to augmented base loci of log divisors.
33 citations
TL;DR: In this paper, it was shown that a reduced divisor can define an integral affine map from the tropical curve to the complete linear system of genus at least two, and a simpler proof of a theorem of Luo on rank-determining sets of points.
Abstract: Given a divisor $D$ on a tropical curve $\Gamma$, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $|D|$. This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduced-divisor map defines an embedding of the curve into the linear system, and in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduced-divisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $C$ and a divisor $D$ of non-negative rank on $C$, reduced divisors equivalent to $D$ define a morphism from $C$ to the complete linear system $|D|$, which is described in terms of Wronskians.
29 citations
TL;DR: In this paper, the Zsigmondy set Z(X,f,P,D) of the sequence defined by the arithmetic intersection of the f-orbit of P with D was studied.
TL;DR: It is shown that an n×n Latin square has at most nO(log k) subsquares of order k and admits at mostN(log n) autotopisms and the theorem by McKay and Wanless that gave a factorial divisor of Rn is extended.
Abstract: A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (?, s, ?) such that L is unchanged after the rows are permuted by ?, the columns are permuted by s and the symbols are permuted by ?. Let n!(n?1)!R n be the number of n×n Latin squares. We show that an n×n Latin square has at most n O(log k) subsquares of order k and admits at most n O(log n) autotopisms. This enables us to show that {ie11-1} divides R n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R n , and give a new proof that R p ?1 (mod p) for prime p.
Posted Content•
TL;DR: In this paper, it was shown that if D is a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p, then each Kahler class on X\D contains a unique asymptotic conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-\epsilon}) if X is smooth.
Abstract: Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \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, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-\epsilon}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce new methods to prove an analogous statement (with rate O(r^{-0.0128})) when X = Bl_{p}P^3 and D = Bl_{p_1,p_2}P^2 is the strict transform of a smooth quadric through p in P^3. Here D is no longer Kahler-Einstein, but the normal 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.
Posted Content•
TL;DR: In this paper, a general theorem concerning the existence and regularity of non-singular compact Kahler manifold with conic singularities along a normal crossing divisor was obtained.
Abstract: Let $X$ be a non-singular compact Kahler manifold, endowed with an effective divisor $D= \sum (1-\beta_k) Y_k$ having simple normal crossing support, and satisfying $\beta_k \in (0,1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X, D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work \cite{CGP} concerning the Monge-Ampere equations on $(X, D)$ by establishing Laplacian and ${\mathscr C}^{2,\alpha, \beta}$ estimates for the solution of this equations regardless to the size of the coefficients $0<\beta_k< 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kahler-Einstein metrics with conic singularities along a normal crossing divisor.
TL;DR: In this paper, the authors constructed a composition of blowings-up such that the transformed pair is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D).
Abstract: Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at \({a \in X}\) if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up \({f:\tilde{X}\rightarrow {X}}\) such that the transformed pair \({(\tilde{X}, \tilde{D})}\) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollar.
Journal Article•
TL;DR: In this article, it was shown that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20.
Abstract: We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.
TL;DR: In this article, the authors presented a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an "integral" version of this result which is of independent interest.
Abstract: Let $\Gamma$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an "integral" version of this result which is of independent interest. As an application, we provide a "geometric proof" of (a dual version of) Kirchhoff's celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${\rm Pic}^g(\Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${\rm Pic}^g(\Gamma)$ is the sum of the volumes of the cells in the decomposition.
TL;DR: In this article, it was shown that #X(K) 6 #X (Fp) + 2r is a good bound for the case of bad reduction, and the new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Cliord's theorem.
Abstract: Let X be a curve over a number eld K with genus g > 2, p a prime of OK over an unramied rational prime p > 2r, J the Jacobian of X, r = rankJ(K), and X a regular proper model of X at p. Suppose r < g. We prove that #X(K) 6 #X (Fp) + 2r, extending the rened version of the Chabauty{Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Cliord’s theorem.
Posted Content•
TL;DR: In this article, Castravet and Tevelev this article studied effective divisors on Ω(n, n) -overline{M}_{0,n}, focusing on hypertree divisor classes.
Abstract: We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a database of hypertree divisor classes and closed formulas for Chen--Coskun divisor classes. We relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n \geq 7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet and Tevelev.
TL;DR: This paper constructs a new class of balanced NPN mappings from Z(p2-p) to Zp for an odd prime p, and presents a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping.
Abstract: A mapping from ZN to ZM can be directly applied for the design of a sequence of period N with alphabet size M, where ZN denotes the ring of integers modulo N. The nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence. When M is a divisor of N, the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation, but it is not balanced. In this paper, we study balanced near-perfect nonlinear (NPN) mappings applicable for the design of sequence sets with low correlation. We first construct a new class of balanced NPN mappings from Z(p2-p) to Zp for an odd prime p. We then present a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping. By applying it to the new class, we obtain a new optimal FHS set of period p2-p with respect to the Peng-Fan bound, whose FHSs are balanced and optimal with respect to the Lempel-Greenberger bound. Moreover, we construct a low-correlation sequence set with size p, period p2-p, and maximum correlation magnitude p from the new class of balanced NPN mappings, which is asymptotically optimal with respect to the Welch bound.
22 May 2013
TL;DR: In this paper, the Kaplansky condition for a K-Hermite ring to be an elementary divisor ring (EDR) was extended to a commutative version.
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.
Posted Content•
TL;DR: In this paper, the authors studied the asymptotic behavior of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor.
Abstract: We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor $Y$. Assuming the data in question is invariant under an $S^1$-action (locally around $Y$) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up. Moreover we recover the existence of the "forbidden region" $R$ on which the density function is exponentially small, and prove that it has an "error-function" behaviour across the boundary $\partial R$. As an illustrative application, we use this to study a certain natural function that can be associated to a divisor in a Kahler manifold.
TL;DR: In this article, the authors define and study the stack of projective projective curves of arithmetic genus g with g smooth marked points forming an ample non-special divisor and give explicit equations of the universal curve (away from characteristics 2 and 3), which can be viewed as a generalization of the Weierstrass cubic and the j-invariant of an elliptic curve.
Abstract: We define and study the stack ${\mathcal U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus g with g smooth marked points forming an ample non-special divisor. We define an explicit closed embedding of a natural ${\mathbb G}_m^g$-torsor over ${\mathcal U}^{ns,a}_{g,g}$ into an affine space and give explicit equations of the universal curve (away from characteristics 2 and 3). This construction can be viewed as a generalization of the Weierstrass cubic and the j-invariant of an elliptic curve to the case g>1. Our main result is that in characteristics different from 2 and 3 our moduli space of non-special curves is isomorphic to the moduli space of minimal A-infinity structures on a certain finite-dimensional graded associative algebra $E_g$ (introduced in arXiv:1208.6332). We show how to compute explicitly the A-infinity structure associated with a curve $(C,p_1,...,p_g)$ in terms of certain canonical generators of the algebra of functions on $C-\{p_1,...,p_g\}$ and canonical formal parameters at the marked points. We study the GIT quotients associated with our representation of ${\mathcal U}^{ns,a}_{g,g}$ as the quotient of an affine scheme by ${\mathbb G}_m^g$ and show that some of the corresponding stack quotients give modular compactifications of ${\mathcal M}_{g,g}$ in the sense of arXiv:0902.3690. We also consider an analogous picture for curves of arithmetic genus 0 with n marked points which gives a new presentation of the moduli space of $\psi$-stable curves (also known as Boggi-stable curves) and its interpretation in terms of $A_\infty$-structures.
Journal Article•
01 Jan 2013-International Journal of Advanced Research in Electrical, Electronics and Instrumentation Energy
TL;DR: The digit recurrence non restoring division algorithm is designed using high speed subtractor and adder and implemented on FPGA xc3s100e-5vq100.
Abstract: Binary division is basically a procedure to determine how many times the divisor D divides the dividend B thus resulting in the quotient Q. At each step in the process the divisor D either divides B into a group of bits or it does not. The divisor divides a group of bits when the divisor has a value less than or equal to the value of those bits. Therefore, the quotient is either 1 or 0. The division algorithm performs either an addition or subtraction based on the signs of the divisor and the partial remainder. There are number of binary division algorithm like Digit Recurrence Algorithm restoring, non-restoring and SRT Division (Sweeney, Robertson, and Tocher), Multiplicative Algorithm, Approximation Algorithms, CORDIC Algorithm and Continued Product Algorithm. This paper focus on the digit recurrence non restoring division algorithm, Non restoring division algorithm is designed using high speed subtractor and adder. High speed adder and subtractor are used to speed up the operation of division. Designing of this division algorithm is done by using VHDL and simulated using Xilinx ISE 8.1i software has been used and implemented on FPGA xc3s100e-5vq100.
TL;DR: The main goal of as mentioned in this paper is to give a characterization of a divisor on an algebraic torus whose complement is Kobayashi hyperbolically imbedded into a toric projective variety.
Abstract: Our main goal of this article is to give a characterization of an algebraic divisor on an algebraic torus whose complement is Kobayashi hyperbolically imbedded into a toric projective variety. As an application of our main theorem, we prove the following: the complement of the union of n + 1 hyperplanes in the n-dimensional projective space \({\mathbb{P}^{n}(\mathbb{C})}\) in general position and a general hypersurface of degree n in \({\mathbb{P}^n(\mathbb{C})}\) is Kobayashi hyperbolically imbedded into \({\mathbb{P}^n(\mathbb{C})}\).
TL;DR: In this article, the authors derived the TD hierarchy associated with a 2 × 2 matrix spectral problem and established Dubrovin-type equation in terms of the introduced elliptic variables, based on the theory of algebraic curve.
Abstract: Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2 × 2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function ϕ, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve \(\mathcal {K}_{n}\). The known zeros and poles of ϕ allow to find theta function representations for ϕ by referring to Riemann’s vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of ϕ and its theta function representation.
Posted Content•
TL;DR: In this paper, the authors studied the asymptotics of the N\'eron height pairing between degree-zero divisors on a family of degenerating compact Riemann surfaces parametrized by an algebraic curve.
Abstract: The aim of this paper is twofold. First, we study the asymptotics of the N\'eron height pairing between degree-zero divisors on a family of degenerating compact Riemann surfaces parametrized by an algebraic curve. We show that if the monodromy is unipotent the leading term of the asymptotics is controlled by the local non-archimedean N\'eron height pairing on the generic fiber of the family. Second, we prove a conjecture of R. Hain to the effect that the `height jumping divisor' related to the normal function $(2g-2)x-K$ on the moduli space $\mm_{g,1}$ of 1-pointed curves of genus $g \geq 2$ is effective. Both results follow from a study of the degeneration of the canonical metric on the Poincar\'e bundle on a family of jacobian varieties.
TL;DR: In this paper, the sum of the sth powers of the positive divisors of a positive integer N was shown to be a function of the number of positive integers in the positive integer n.
Abstract: Let s(N) denote the sum of the sth powers of the positive divisors of a positive integer N and let e s(N) = P djN ( 1) d 1 d s with d, N, and s positive integers. Hahn (12) proved that
TL;DR: In this article, a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL3 on a smooth, marked curve (C, $$ \vec{p} $$¯¯¯¯ ): Elements of this algebra have a well known interpretation as conformal blocks.
Abstract: We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL3(
$$ \mathbb{C} $$
) bundles on a smooth, marked curve (C, $$ \vec{p} $$
): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $$ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $$
of quasi-parabolic principal SL3(
$$ \mathbb{C} $$
) bundles on (C, $$ \vec{p} $$
). Along the way we recover positive polyhedral rules for counting conformal blocks.
TL;DR: Droll's result is extended by drawing up a complete list of all graphs having the Ramanujan property for each prime power $p^s$ and arbitrary divisor set ${\cal D}$.
Abstract: A connected $\rho$-regular graph $G$ has largest eigenvalue $\rho$ in modulus. $G$ is called Ramanujan if it has at least $3$ vertices and the second largest modulus of its eigenvalues is at most $2\sqrt{\rho-1}$. In 2010 Droll classified all Ramanujan unitary Cayley graphs. These graphs of type ${\rm ICG}(n,\{1\})$ form a subset of the class of integral circulant graphs ${\rm ICG}(n,{\cal D})$, which can be characterised by their order $n$ and a set $\cal D$ of positive divisors of $n$ in such a way that they have vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{(a,b):\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in {\cal D}\}$. We extend Droll's result by drawing up a complete list of all graphs ${\rm ICG}(p^s,{\cal D})$ having the Ramanujan property for each prime power $p^s$ and arbitrary divisor set ${\cal D}$.