Showing papers on "Divisor published in 2019"

Wall crossing for K-moduli spaces of plane curves

Abstract: We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as $c$ varies. The main application in this paper is the case of plane curves of degree $d \geq 4$ as boundary divisors of $\mathbb{P}^2$. In this case, we show that when the coefficient $c$ is small, the K-moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K-moduli spaces are weighted blow-ups of Kirwan type. We also describe all wall crossings for degree 4,5,6, and relate the final K-moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

A Survey on the Computation of Quaternions From Rotation Matrices

Abstract: The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, 3D image processing, computer graphics, etc. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in $\R^3$, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to $3{\times}3$ rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from $4{\times}4$ rotation matrices, is the most robust method when particularized to three dimensions.

Reheating and dark radiation after fibre inflation

Abstract: We study perturbative reheating at the end of fibre inflation where the inflaton is a closed string modulus with a Starobinsky-like potential. We first derive the spectral index $n_s$ and the tensor-to-scalar ratio $r$ as a function of the number of efoldings and the parameter $R$ which controls slow-roll breaking corrections. We then compute the inflaton couplings and decay rates into ultra-light bulk axions and visible sector fields on D7-branes wrapping the inflaton divisor. This leads to a reheating temperature of order $10^{10}$ GeV which requires $52$ efoldings. Ultra-light axions contribute to dark radiation even if $\Delta N_{\rm eff}$ is almost negligible in the generic case where the visible sector D7-stack supports a non-zero gauge flux. If the parameter $R$ is chosen to be small enough, $n_s\simeq 0.965$ is then in perfect agreement with current observations while $r$ turns out to be of order $r\simeq 0.007$. If instead the flux on the inflaton divisor is turned off, $\Delta N_{\rm eff}\lesssim 0.6$ which, when used as a prior for Planck data, requires $n_s\simeq 0.99$. After $R$ is fixed to obtain such a value of $n_s$, primordial gravity waves are larger since $r\simeq 0.01$.

Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges

Abstract: We study the problem of obtaining asymptotic formulas for the sums $$\sum _{X < n \le 2X} d_k(n) d_l(n+h)$$ and $$\sum _{X < n \le 2X} \Lambda (n) d_k(n+h)$$ , where $$\Lambda $$ is the von Mangoldt function, $$d_k$$ is the $$k^{{\text {th}}}$$ divisor function, X is large and $$k \ge l \ge 2$$ are integers. We show that for almost all $$h \in [-H, H]$$ with $$H = (\log X)^{10000 k \log k}$$ , the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $$\Lambda (n) \Lambda (n + h)$$ and we obtained better estimates for the error terms at the price of having to take $$H = X^{8/33 + \varepsilon }$$ .

The functional graph of linear maps over finite fields and applications

Abstract: Let $$\mathbb F_{q}$$ be the finite field with q elements and $$n\ge 2$$ be a positive integer. We study the functional graph associated to linear maps over finite fields. In particular, we describe the functional graph $$\mathcal {G}_f(\mathbb F_{q^n})$$ associated to the map induced by $$L_f$$ on $$\mathbb F_{q^n}$$ , where f is any irreducible divisor of $$x^n-1$$ over $$\mathbb F_q$$ and $$L_f$$ is the q-associate of f. This description derives interesting information on the graph $$\mathcal {G}_f(\mathbb F_{q^n})$$ , such as the number of cycles and the average of the preperiod length. When $$\gcd (f, x^n-1)=1$$ , $$L_f$$ is a permutation on $$\mathbb F_{q^n}$$ and the cycle decomposition of $$\mathcal {G}_f(\mathbb F_{q^n})$$ is well known. In this case, we present some applications of this result, such as the construction of linear involutions over odd characteristic and permutations with few fixed points.

Reducible M -curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

Abstract: We associate real and regular algebraic–geometric data to each multi-line soliton solution of Kadomtsev–Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non-negative part of real Grassmannians $$Gr^{\mathrm{TNN}}(k,n)$$ . In [3] we were able to construct real algebraic–geometric data for soliton data in the main cell $$Gr^{\mathrm{TP}} (k,n)$$ only. Here we do not just extend that construction to all points in $$Gr^{\mathrm{TNN}}(k,n)$$ , but we also considerably simplify it, since both the reducible rational $$\texttt {M}$$ -curve $$\Gamma $$ and the real regular KP divisor on $$\Gamma $$ are directly related to the parametrization of positroid cells in $$Gr^{\mathrm{TNN}}(k,n)$$ via the Le-networks introduced in [63]. In particular, the direct relation of our construction to the Le-networks guarantees that the genus of the underlying smooth $$\texttt {M}$$ -curve is minimal and it coincides with the dimension of the positroid cell in $$Gr^{\mathrm{TNN}}(k,n)$$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $$Gr^{\mathrm{TP}}(2,4)$$ and we compare it with that in [3].

Remarks on special kinds of the relative log minimal model program

Abstract: We prove $$\mathbb {R}$$ -boundary divisor versions of results proved by Birkar (Publ Math Inst Hautes Etudes Sci 115(1):325–368, 2012) and Hacon–Xu (Invent Math 192(1):161–195, 2013) on special kinds of the relative log minimal model program.

S-duality and refined BPS indices

Abstract: Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $\mathbb{P}^2$, and make them explicit for $U(4)$.

Projective klt pairs with nef anti-canonical divisor

Abstract: In this paper, we study a projective klt pair $(X, \Delta)$ with the nef anti-log canonical divisor $-(K_X+\Delta)$ and its maximally rationally connected fibration $\psi: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+\Delta)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+\Delta_{X_y})$ on a general fiber $X_y$ of $\psi: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness, which gives a sharper estimate than the question posed by Hacon-$\mathrm{M^{c}}$Kernan. Moreover, in the case of $X$ being smooth, we show that a maximally rationally connected fibration $\psi: X \to Y$ can be chosen to be a morphism to a smooth projective variety $Y$ with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair $(X, \Delta)$, which can be seen as a generalization of Cao-Horing's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair $(X, \Delta)$ with $-(K_X +\Delta)$ nef, and obtain a structure theorem for projective orbifold surfaces.

Mirror theorems for root stacks and relative pairs

Abstract: Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental’s Lagrangian cone, for Gromov–Witten theory of the root stack $$X_{D,r}$$ . As an application, we also obtain an I-function for relative Gromov–Witten theory following the relation between relative and orbifold Gromov–Witten invariants.

On the log-local principle for the toric boundary

Abstract: Let $X$ be a smooth projective complex variety and let $D=D_1+\cdots+D_l$ be a reduced normal crossing divisor on $X$ with each component $D_j$ smooth, irreducible, and nef. The log-local principle of van Garrel-Graber-Ruddat conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of $(X,D)$ is equivalent to the genus 0 local Gromov-Witten theory of $X$ twisted by $\bigoplus_{j=1}^l\mathcal{O}(-D_j)$. We prove that an extension of the log-local principle holds for $X$ a (not necessarily smooth) $\mathbb{Q}$-factorial projective toric variety, $D$ the toric boundary, and descendent point insertions.

Special Lagrangian submanifolds of log Calabi-Yau manifolds

Abstract: We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that $X$ admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that $Y$ is a rational elliptic surface, or $Y= \mathbb{P}^2$ we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, $X$ can be compactified to the complement of a Kodaira type $I_{d}$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.

A High-Speed Division Algorithm for Modular Numbers Based on the Chinese Remainder Theorem with Fractions and Its Hardware Implementation

Abstract: In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea of the algorithm is (a) to transform the residual representation of the dividend and divisor into a weighted fixed-point code and (b) to find the higher power of 2 in the divisor written in a residue number system (RNS). This information is acquired using the CRT with fractions: higher power is defined by the number of zeros standing before the first significant digit. All intermediate calculations of the algorithm involve the operations of right shift and subtraction, which explains its good performance. Due to the abovementioned techniques, the algorithm has higher speed and consumes less computational resources, thereby being more appropriate for the multidigit division of modular numbers than the algorithms described earlier. The new algorithm suggested in this paper has O (log2 Q) iterations, where Q is the quotient. For multidigit numbers, its modular division complexity is Q(N), where N denotes the number of bits in a certain fraction required to restore the number by remainders. Since the number N is written in a weighed system, the subtraction-based comparison runs very fast. Hence, this algorithm might be the best currently available.

Formulas for Chebotarev densities of Galois extensions of number fields

Abstract: We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if \(C \subset G = {{\mathrm{Gal}}}(L/K)\) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K: $$\begin{aligned} -\lim _{X\rightarrow \infty } \sum _{\begin{array}{c} 2\le N(I)\le X \\ I \in S(L/K; C) \end{array}} \frac{\mu _K(I)}{N(I)} = \frac{|C|}{|G|}, \end{aligned}$$ where \(\mu _K(I)\) denotes the generalized Mobius function and S(L / K; C) is the set of ideals \(I\subset \mathcal {O}_K\) such that I has a unique prime divisor \(\mathfrak {p}\) of minimal norm and the Artin symbol \(\left[ \frac{L/K}{\mathfrak {p}}\right] \) is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.

A complete characterization of pretty good state transfer on paths

Abstract: We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If $n$ is the length of the path, and the vertices are indexed by the positive integers from 1 to $n$, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices $a$ and $b$ are that (a) $a + b = n + 1$, (b) $n + 1$ has at most one odd non-trivial divisor, and (c) if $n = 2^t r - 1$, for $r$ odd and $r eq 1$, then $a$ is a multiple of $2^{t - 1}$.

Equivariant quantum cohomology of the odd symplectic Grassmannian

Abstract: The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $$\mathrm {IG}$$ with two orbits, and $$\mathrm {IG}$$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $$\mathrm {IG}(k, 2n+2)$$ . We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $$\mathrm {IG}$$ , i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case $$k=2$$ , and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.

Mukai models and Borcherds products

Abstract: Let F_{g,n} be the moduli space of n-pointed K3 surfaces of genus g with at worst rational double points. We prove that the ring of pluricanonical forms on F_{g,n} is isomorphic to the ring of orthogonal modular forms of weight divisible by 19+n, twisted by the determinant character and with vanishing condition at the (-2)-Heegner divisor. This maps canonical forms on a smooth projective model to cusp forms. Then we use Borcherds products to find a lower bound of n where F_{g,n} has nonnegative Kodaira dimension, and compare this with an upper bound where F_{g,n} is unirational or uniruled using classical and Mukai models in g<21. In some cases, this reveals the exact transition point of Kodaira dimension. When n is sufficiently large, the Kodaira dimension stabilizes to 19.

Characterizations of toric varieties via polarized endomorphisms

Abstract: Let X be a normal projective variety and $$f:X\rightarrow X$$ a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is $$\mathbb {Q}$$ -factorial and G-almost homogeneous for some linear algebraic group G such that f is G-equivariant, then X is a toric variety. Next we give a geometric characterization: if X is of Fano type and smooth in codimension 2 and if there is an $$f^{-1}$$ -invariant reduced divisor D such that $$f|_{X\backslash D}$$ is quasi-etale and $$K_X+D$$ is $$\mathbb {Q}$$ -Cartier, then X admits a quasi-etale cover $${\widetilde{X}}$$ such that $${\widetilde{X}}$$ is a toric variety and f lifts to $${\widetilde{X}}$$ . In particular, if X is further assumed to be smooth, then X is a toric variety.

Lie Polynomials and a Twistorial Correspondence for Amplitudes

Abstract: We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $\mathcal{M}_{0,n}$, the moduli space of $n$ points on the Riemann sphere up to Mobius transformation. We introduce a twistorial correspondence between the cotangent bundle $T^*_D\mathcal{M}_{0,n}$, the bundle of forms with logarithmic singularities on the divisor $D$ as the twistor space, and $\mathcal{K}_n$ the space of momentum invariants of $n$ massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain $n-3$-forms on $\mathcal{K}_n$, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $n-3$-planes in $\mathcal{K}_n$ introduced by ABHY.}

On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Abstract: Let $N$ be a non-squarefree positive integer and let $\ell$ be an odd prime such that $\ell^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for $\Gamma_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\ell$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] eq 0$, where $\mathcal{C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.

Two or Few-Weight Trace Codes over ${\mathbb{F}_{q}}+u{\mathbb{F}_{q}}$

Abstract: Let $p$ be a prime number and $q=p^{s}$ for a positive integer $s$ . For any positive divisor $e$ of $q-1$ , we construct infinite families of codes $\mathcal {C}$ of size $q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the ring $R= \mathbb {F}_{q} + u \mathbb {F}_{q}$ , $u^{2} = 0$ . Using Gaussian sums, their Lee weight distributions are provided. In particular, when $\gcd (e,m)=1$ , under the Gray map, the images of all codes in $\mathcal {C}$ are of two-weight over the finite field $\mathbb {F}_{q}$ , which meet the Griesmer bound. Moreover, when $\gcd (e,m)=2, 3$ , or 4, all codes in $\mathcal {C}$ are of most five-weight codes.

Nef anti-canonical divisors and rationally connected fibrations

Abstract: We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.

Torus fibers and the weight filtration

Abstract: We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus Y;\mathbb{Q})$ for all $k$. We show that an analogous result holds for degenerations of Calabi--Yau varieties. We use this to show that P=W type results hold for pairs $(X,Y)$ consisting of a rational surface $X$ and a nodal anticanonical divisor $Y$, and for K3 surfaces.

The rational cuspidal divisor class group of $X_0(N)$.

Abstract: For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group $\mathcal{C}(N)$ of $X_0(N)$, which is conjecturally equal to the group of rational torsion points on $J_0(N)$. More specifically, let $\ell$ be any given prime. For a non-trivial divisor $d$ of $N$, we construct a rational cuspidal divisor $Z_\ell(d)$ and show that the $\ell$-primary subgroup of $\mathcal{C}(N)$ is isomorphic to the direct sum of the cyclic groups generated by the images of the divisors $Z_\ell(d)$. Also, we compute the order of the image of the divisor $Z_\ell(d)$ in $J_0(N)$.

Tau functions, Prym-Tyurin classes and loci of degenerate differentials

Abstract: We study the rational Picard group of the projectivized moduli space $$P\overline{{\mathfrak {M}}}_{g}^{(n)}$$ of holomorphic n-differentials on complex genus g stable curves. We define $$n-1$$ natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve.

An improved sieve of Eratosthenes

Abstract: We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and essentially linear time. These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and essentially linear time. We can also apply our sieve to any subinterval of [1,N] of length O(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length >> sqrt{N}. Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ties to Voronoi's work on the Dirichlet divisor problem. The advantage of the method here resides in the fact that, because the method we will give is based on the sieve of Eratosthenes, we will also be able to use it to factor integers, and not just to produce lists of consecutive primes.

The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants

Abstract: This paper deals with the invariant $$R_X$$ called the RR-correction term, which appears in the Riemann–Roch and Numerical Adjunction Formulas for normal surface singularities. Typically, $$R_X=\delta ^\text {top}_X-\delta ^\text {an}_X$$ decomposes as difference of topological and analytical local invariants of its singularities. The invariant $$\delta ^\text {top}_X$$ is well understood and depends only on the dual graph of a good resolution. The purpose of this paper is to give a new interpretation for $$\delta ^\text {an}_X$$ , which in the case of cyclic quotient singularities can be explicitly computed via generic divisors. We also include two types of applications: one is related to the McKay decomposition of reflexive modules in terms of special reflexive modules in the context of the McKay correspondence. The other application answers two questions posed by Blache (Abh Math Semin Univ Hambg 65:307–340, 1995) on the asymptotic behavior of the invariant $$R_X$$ of the pluricanonical divisor.

Group divisible designs with block size four and type gum1—III

Abstract: We show that the necessary conditions for the existence of group divisible designs with block size four (4‐GDDs) of type gum1 are sufficient for g ≡ 0 (mod h), h = 39, 51, 57,69, 87, 93, 111, 123 and 129, and for g = 13, 17, 19, 23, 25,29, 31 and 35. More generally, we show that for g ≡ 3 (mod 6), the possible exceptions occur only when u = 8, and there are no exceptions at all if g/3 has a divisor d>1 such that d ≡ 1 (mod 4) or d is a prime not greater than 43. Hence there are no exceptions when g ≡ 3 (mod 12). Consequently, we are able to extend the known spectrum for g ≡ 1 and 5 (mod 6). Also, we complete the spectrum for 4‐GDDs of type (3α)4(6α)1(3b)1.

Second Chern class of Fano manifolds and anti-canonical geometry

Abstract: Let X be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of X in terms of its index and degree. As an application, if Y is a n-dimensional Fano manifold with $$-K_Y=(n-3)H$$ for some ample divisor H, we prove that $$h^0(Y,H)\ge n-2$$ . Moreover, we show that the rational map defined by $$\vert mH\vert $$ is birational for $$m\ge 5$$ , and the linear system $$\vert mH\vert $$ is basepoint free for $$m\ge 7$$ . As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most 4 are also investigated.