Field seminar "Flexibility and Computational Methods II"

In this talk, we delve into examining extensions for specific non-special curves of genus 5 in P^r. Initially, we establish a classification for linearly normal surfaces S embedded in P^(d-4) with sectional genus 5 and degree d not less than 9. This classification serves as the foundation for our exploration of extension theory, which includes investigating general curves, trigonal curves of genus 5, and the presence of sectional genus 5 manifolds. Moreover, we present two new examples of smooth Calabi-Yau threefolds in P^7 of degree 19. 

Presentation

Abstract. We discuss the similarities and differences between linear algebraic groups and automorphism groups of algebraic varieties. We are especially interested in the group of special automorphisms SAut(X), i.e., the normal subgroup of Aut(X) generated by all one-parameter additive subgroups. We focus on the flexibility property for affine varieties. The group SAut(X) acts in the regular locus of a flexible variety X infinitely transitively, that is, any finite collection of smooth points can be sent to any finite collection of smooth points of the same cardinality. Using flexibility, we show that every non-degenerate toric variety, every homogeneous space of a semisimple group, and every variety covered by affine spaces admits a surjective morphism from an affine space. Applying the ellipticity property introduced by Mikhail Gromov in 1989, we prove that a complete algebraic variety X is an image of an affine space if and only if X is unirational. This is a joint result with Shulim Kaliman and Mikhail Zaidenberg. Also we obtain an analog of this result for affine cones. 

Presentation

In this talk, we provide a smooth family of complete intersections of three quadrics in P^7 with both fibers with flexible affine cones and fibers with no flexible ones. 

Lecture notes

Flexibility of an affine cone can be expressed in terms of cylinders on the underlying projective variety. We will survey known cylinder constructions for del Pezzo surfaces and weak del Pezzo surfaces. We will also present a unified description of cylinders for surfaces of varying degree. Our description uses relation between bubble cycles and base schemes of pencils of curves on the projective plane.

Presentation

Presentation

Rational curves play important role in the geometry of Fano manifolds. Minimal rational curves (i.e., those which do not degenerate to non-reduced or reducible curves) are similar (to a certain extent) to geodesics on Riemannian manifolds. However, unlike geodesics, in general one cannot draw a minimal rational curve at an arbitrary direction through a general point. Admissible directions form the variety of minimal rational tangents (VMRT) in the projectivized tangent space at a general point. It follows from works of J.-M. Hwang, N. Mok, and others that the geometry of Fano manifolds with Picard number 1 is largely controlled by their VMRT. In certain cases one can recover a Fano manifold from its VMRT via a differential-geometric structure, whose starting point is the Lie algebra of symbols of the filtered system of distributions in the tangent bundle generated by VMRT. Such a Lie algebra can be constructed from an arbitrary embedded projective variety. It deserves to be studied as a new algebraic invariant of projective varieties. We shall discuss these issues in the talk.

Presentation

We survey results on open embeddings of the affine space A^n into a complete algebraic variety X such that the action of the vector group G on A^n by translations extends to an action of G on X. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of A^n into projective spaces and give its generalization for embeddings into projective hypersurfaces. It is proved that non-degenerate projective hypersurfaces admitting such an embedding are in bijection with Gorenstein local algebras. Moreover, such an embedding into a projective hypersurface is unique if and only if the hypersurface is non-degenerate. Further we mention embeddings into complete toric varieties, flag varieties and their degenerations, and Fano varieties of certain types.

Presentation

This talk provides a classification of additive actions on P^n with small modality.

Lecture notes

In the study of automorphism groups of toric varieties, a key role is played by one-parameter additive groups normalized by the acting torus. Such subgroups are called root subgroups and each of them is uniquely determined by its weight, called a Demazure root of the corresponding toric variety. Moreover, the set of all Demazure roots admits an explicit combinatorial description in terms of the fan defining the toric variety. In the setting of arbitrary connected reductive groups acting on algebraic varieties, a natural generalization of toric varieties is given by spherical varieties. A spherical variety is an algebraic variety X equipped with an action of a connected reductive group G in such a way that a Borel subgroup B of G has a dense open orbit in X. A proper generalization of root subgroups for spherical varieties is given by one-parameter additive groups normalized by B, which we call B-root subgroups. In the talk we plan to discuss B-root subgroups on spherical varieties, including simples properties, applications, and open problems. The talk is based on joint works of the speaker with I. Arzhantsev and with V. Zhgoon.

Presentation

Let G be a linear algebraic group acting on an affine variety V, all are defined over a valued field k. We may endow G(k) and V(k) with the v-adic topology induced from that of the base field k. The rational orbit G(k).v of a rational point v is called cocharacter closed (resp., Hausdorff closed) if this orbit contains the limit point (if exists) along any cocharacter of G (resp., this orbit is closed relative to the above v-adic topology). In 2005, G. Roehrle et al proposed a geometric approach to study completely reducible subgroups due to J.-P. Serre via cocharacter closedness of rational orbits. The aim of this talk is to discuss the relationship between the cocharacter closedness and Hausdorff closedness of rational orbits, as well as the cocharacter closure and Hausdorff closure of these ones. This is a joint work with Vu Tuan Hien and Vo Duy Hoang.

Presentation

Suppose an algebraic group G acts on an affine variety X with an open orbit and the stabilazer of any point contains a maximal unipotent subrgoup of G. Such varieties are called horospherical. In 1972 V .Popov and E. Vinberg obtained a combinatorical description of this class of varieties, which is a generalization of a well-known one for toric varieties. In 2017 A. Shafarevich proved that a horospherical variety is flexible if G is semisimple. But if G is a reductive group, that is not semisimple and X is not normal, the variety X can be non-flexible. In this talk we plan to prove that a normal horospherical variety with only constant invertible functions is flexible. Also we discuss orbits of automorphism group on this varieties. The talk is based on joint works with V. Borovik, I. Boldyrev, and A. Shafarevich.

Presentation

Let R = K[x_1,...,x_n] be a polynomial ring over a field K. Let G=(V,E) be a graph with V = {x_1,...,x_n}. The edge ideal of G is defined by I(G) =(x_i x_j | x_i x_j is in E) \subset R. G is a Gorenstein graph if R/I(G) is a Gorenstein ring. In this talk, we focus on the following problem: characterize Gorenstein graphs G in terms of combinatorial properties of G.

Lecture notes