We describe two geometrically meaningful compactifications of the moduli space elliptic K3 surfaces via stable slc pairs, for different choices a polarizing divisor, and show that their normalizations are toroidal space, one ramification divisor another rational curve divisor. In course proof, we further develop theory integral affine spheres with 24 singularities. also construct (generalized) ...

In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has

let $m$ be an $r$-module and $0 neq fin m^*={rm hom}(m,r)$. we associate an undirected graph $gf$ to $m$ in which non-zero elements $x$ and $y$ of $m$ are adjacent provided that $xf(y)=0$ or $yf(x)=0$. weobserve that over a commutative ring $r$, $gf$ is connected anddiam$(gf)leq 3$. moreover, if $gamma (m)$ contains a cycle,then $mbox{gr}(gf)leq 4$. furthermore if $|gf|geq 1$, then$gf$ is finit...

We prove that if the Logarithmic Comparison Theorem holds for a free divisor in a complex manifold then this divisor is Euler homogeneous. F.J. Calderón–Moreno et al. have conjectured this statement and have proved it for reduced plane curves.

The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33

In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length n over a finite field Fq, in the case which each prime divisor of n is also a divisor of q − 1.

We prove a non-vanishing theorem of the cohomology H0 of the adjoint divisor KX + dLe where dLe is the round up of a nef and big Q-divisor L.

The extended greatest common divisor GCD problem is given a vector a a an of positive integers compute g the greatest common divisor of these inte gers and nd a vector x x xn of integer coe cients such that

Abstract We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is deformation complement simple crossings divisor. also rigidity results for skeleton divisor complement.