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## Sharygin triangles and elliptic curves

Cornell University
,
2016.

Netay I. V., Савватеев А. В.

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.

Michael Finkelberg, Leonid Rybnikov, / Cornell University. Series math "arxiv.org". 2013.

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic ...

Added: December 27, 2013

Kharlamov V., Viktor Kulikov, / Cornell University. Series math "arxiv.org". 2013.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli ...

Added: December 27, 2013

Brown F., / arxive. Series math "nt". 2013. No. 1110.6917.

Abstract. We study the de Rham fundamental group of the configuration space
E (n) of n + 1 marked points on a complex elliptic curve E, and define multiple
elliptic polylogarithms. These are multivalued functions on E (n) with unipotent
monodromy, and are constructed by a general averaging procedure. We show
that all iterated integrals on E (n) , ...

Added: May 14, 2014

Bezrukavnikov R., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Lee K., Shabalin T., / Cornell University. Series math "arxiv.org". 2014.

We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements. ...

Added: October 17, 2014

Lev Soukhanov, / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

Ivan Cheltsov, Park J., Won J., / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Takebe T., Kuroki G., Journal of Physics A: Mathematical and Theoretical 2001 Vol. 34 No. 11 P. 2403-2413

We construct a Gaudin type lattice model as the Wess-Zumino-Witten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique. ...

Added: August 14, 2014

Rybakov S., / Cornell University. Series math "arxiv.org". 2014.

A k-isogeny class of abelian varieties over a finite field k is uniquely determined by the Weil polynomial f of any variety from this class. When we consider classification problems concerning abelian varieties inside an isogeny class, the classification can be given in terms of the corresponding Weil polynomial. In this paper we improve our ...

Added: January 21, 2014

Fedor Bogomolov, Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2013.

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases. ...

Added: November 21, 2014

Fonarev A., Moscow Mathematical Journal 2016 Vol. 16 No. 4 P. 711-726

We classify irreducible equivariant Ulrich vector bundles on isotropic Grassmannians. ...

Added: November 7, 2017

Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...

Added: February 6, 2013

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2012. No. 1210.3339.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...

Added: September 14, 2013

Arzhantsev I., Hausen J., Herppich E. et al., Moscow Mathematical Journal 2014 Vol. 14 No. 3 P. 429-471

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result which, for example, settles the case of a finitely generated algebra of ...

Added: November 15, 2013

F.A. Bogomolov, Vik.S. Kulikov, / Cornell University. Series math "arxiv.org". 2014.

In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each
equipped finite group $(G,O)$. It is equal to the number of connected
components of a Hurwitz space parametrizing coverings of a projective line with
Galois group $G$ assuming that all local monodromies belong to conjugacy
classes $O$ in $G$ and the number of branch points is greater than some
constant. ...

Added: November 21, 2014

Campana F., Demailly J., Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2013.

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of ...

Added: May 13, 2013

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Pavlov A., Mathematische Zeitschrift 2021 No. 297 P. 223-254

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve E embedded in P2 as a smooth cubic, we get explicit values for Betti ...

Added: October 31, 2020

Fedor Bogomolov, De Oliveira B., / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Bogomolov F. A., Fu H., European Journal of Mathematics 2018 Vol. 4 No. 2 P. 555-560

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points. ...

Added: September 13, 2018

Victor Kulikov, Shustin E., / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Positselski L., Efimov A., / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

Efimov A., / Cornell University. Series math "arxiv.org". 2013.

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary ...

Added: October 31, 2013