Let $F$ be a field and $Gr\left(i,F^n\right)$ be the Grassmannian of $i$-dimensional linear subspaces of $F^n$. A map $f:Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$ is called nesting if $l\subset f\left(l\right)$ for every $l\in Gr\left(i,F^n\right)$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr\left(i,{\mathbb{C}}^n\right)\to Gr\left(j,{\mathbb{C}}^n\right)$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$, where $F$ is an algebraically closed field of arbitrary characteristic. For $i=1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space $P_F^n$.

The aim of this article is to present five new examples of modular rigid Calabi-Yau threefolds by giving explicit correspondences to newforms of weight 4 and levels 10, 17, 21 and 73.

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety $V$ having a regular action of a finite group $G$. In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture...

The hypersurface in ${ℂ}^3$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_r,r=6,7,8$, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_r$ provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra $ℂ[x_1,x_2,x_3]$ to a noncommutative algebra with generators $x_1,x_2,x_3$ and the following 3 relations labelled...

In this article, we prove that there does not exist a family of maximal rank of entire curves in the universal family of hypersurfaces of degree $d\ge 2n$ in the complex projective space ${\mathbb{P}}^n$. This can be seen as a weak version of the Kobayashi conjecture asserting that a general projective hypersurface of high degree is hyperbolic in the sense of Kobayashi.