Bulletin of the
Korean Mathematical Society BKMS

Magadan Volume

On three-dimensional semi-terminal singularities

Abstract : We classify three-dimensional non-normal semi-terminal singularities.

Birational rigidity is not an open property

Abstract : We show that birational rigidity of Mori fibre spaces is not open in moduli.

Laurent phenomenon for Landau--Ginzburg models of complete intersections in Grassmannians of planes

Abstract : In a spirit of Givental's constructions Batyrev, Ciocan-Fonta\-nine, Kim, and van Straten suggested Landau--Ginzburg models for smoo\-th Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections in complex tori equipped with special functions called superpotentials. We provide a particular algorithm for constructing birational isomorphisms of these models for complete intersections in Grassmannians of planes with complex tori. In this case the superpotentials are given by Laurent polynomials. We study Givental's integrals for Landau--Ginzburg models suggested by Batyrev, Ciocan-Fontanine, Kim, and van Straten and show that they are periods for pencils of fibers of maps provided by Laurent polynomials we obtain. The algorithm we provide after minor modifications can be applied in a more general context.

On contraction of algebraic points

Abstract : We study contraction of points on $\mathbb{P}^1(\bar{\mathbb{Q}})$ with certain control on local ramification indices, with application to the unramified curve correspondence problem initiated by Bogomolov and Tschinkel.

Sharygin triangles and elliptic curves

Abstract : The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles.

A simple proof of the non-rationality of a general quartic double solid

Abstract : The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.

Birationally rigid complete intersections of codimension two

Abstract : We prove that in the parameter space of $M$-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least $\frac12 (M-9)(M-10)-1$.

Cylinders in del Pezzo surfaces with du Val singularities

Abstract : We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders.

On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two

Abstract : The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.