Related papers: The Cayley Plane and the Witten Genus
For every fixed genus $g\geq 1$, we consider all quadruples $Q=(w_0,w_1,w_2,d)\in\mathbb{Z}^4_{>0}$ with the property that any smooth degree-$d$ curve embedded in the weighted projective plane $\mathbb{P}^2(w_0,w_1,w_2)$ has genus $g$. We…
The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a derive general lower bound for the genus of…
We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle $Q$, the degenerate homology of $Q$ is completely determined by the…
In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…
Theorem: If W is a smooth complex projective variety with h^1 (O-script_W) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary: A smooth…
A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to the subset of diffeomorphisms that become physical at corners -- codimension-$2$ surfaces supporting Noether charges. It…
Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold…
This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a…
In this article, we aim to largely complete the program of proving the Tate conjecture for surfaces of geometric genus one, by introducing techniques to analyze those surfaces whose "natural models" are singular. As an application, we show…
By carrying out a rational transformation on the base curve $\mathbb{CP}^1$ of the Seiberg-Witten curve for $\mathcal{N}=2$ supersymmetric pure $\mathrm{SU}(2)$-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard…
Let X be a non-singular algebraic curve of genus at least 3 and let M denote the moduli space of stable vector bundles of rank n and fixed determinant of degree d with n and d coprime. For any semistable bundle E over X, we can pull E back…
A new global approach in the study of duality transformations is introduced. The geometrical structure of complex line bundles is generalized to higher order U(1) bundles which are classified by quantized charges and duality maps are…
Surfaces of general type with geometric genus $p_g=0$, which can be given as Galois covering of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geq 2$ and $q$ is a prime…
The Hartshorne--Hirschowitz theorem says that a generic union of lines in $\mathbb{P}^n$, $(n\geq 3)$, has good postulation. The proof of Hartshorne and Hirschowitz in the initial case $\mathbb{P}^3$ is difficult and so long, which is…
We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$,…
We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of…
We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as…
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
In the generalized Legendre approach, the equation describing an asymptotically locally Euclidean space of type $D_n$ is found to admit an algebraic formulation in terms of the group law on a Weierstrass cubic. This curve has the structure…
In this article we construct a specific projective degeneration of K3 surfaces of degree 2g-2 in P^g to a union of 2g-2 planes, which meet in such a way that the combinatorics of the configuration of planes is a triangulation of the…