Del Pezzo Zoo
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose alpha-invariant of Tian is greater than 2/3.
math.AG↗
arXiv subjects
Constantin Shramov
Publications and source records attributed to Constantin Shramov.
Three embeddings of the Klein simple group into the Cremona group of rank three
We study the action of the Klein simple group G consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.
math.AG↗
Five embeddings of one simple group
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that the Cremona group of rank 3 has at least five non-conjugate subgroups isomorphic to A6.
math.AG↗
Six-dimensional exceptional quotient singularities
We classify six-dimensional exceptional quotient singularities and show that seven-dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irreducible six-dimensional projective representation of the sporadic simple Hall--Janko group gives rise to an exceptional quotient singularity.
math.AG↗
On exceptional quotient singularities
We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the $α$-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient singularities.
math.AG↗
Exceptional del Pezzo hypersurfaces
We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold Kähler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.
math.AG↗
Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly
We compute global log canonical thresholds of some smooth Fano threefolds.
math.AG↗
Extremal metrics on del Pezzo threefolds
We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variety is equal to 1/2.
math.AG↗
Birational automorphisms of nodal quartic threefolds
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete the proof of birational rigidity over a non-closed field and describe the situations when some of the birational involutions in question become regular (and, in particular, complete the proof of the initial theorem on birational rigidity, since some details were not established in the original paper of M.Mella).
math.AG↗
Q-factorial quartic threefolds
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give some geometrical constructions related to the latter quartic.
math.AG↗
Birational rigidity and Q-factoriality of a singular double quadric
We prove birational rigidity and calculate the group of birational automorphisms of a nodal Q-factorial double cover $X$ of a smooth three-dimensional quadric branched over a quartic section. We also prove that $X$ is Q-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-Q-factorial variety of this type with 12 simple double singularities.
math.AG↗
On rationality of nonsingular threefolds with a pencil of Del Pezzo surfaces of degree 4
We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in a scroll over $P^1$. As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over $P^1$: we prove that any fibration of this kind with the topological Euler characteristic -4 is rational.
math.AG↗
Hyperelliptic and trigonal Fano threefolds
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.
math.AG↗