# Aleksandr V. PukhlikovUniversity of Liverpool | UoL · Department of Mathematical Sciences

Aleksandr V. Pukhlikov

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153

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Introduction

Aleksandr V. Pukhlikov currently works at the Department of Mathematical Sciences, University of Liverpool. Aleksandr does research in Algebraic Geometry. Their most recent publication is 'Birationally rigid complete intersections of high codimension'.

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## Publications

Publications (153)

In this note we improve the theorem on Galois rational covers $X\dashrightarrow V$ for primitive Fano varieties $V$, recently proven by the author, in the two directions: we extend to the maximum the class of Galois groups $G$, for which the proof works, and relax the conditions that must be satisfied by the variety $V$ -- the divisorial canonicity...

We show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where \(d\geqslant 4\), is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity con...

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain conditions of general position, under the assumption that the fibration $V/S$ is sufficiently twisted over the ba...

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers $X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group, where $X$ is a rationally connected variety. In particular, there are no rational maps $X\dashrightarrow V$ of degree 2 with $X$...

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is...

In this paper we describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension $\geqslant 8$ with at mostquadratic singularities of rank $\geqslant 8$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected...

In this paper we prove birational superrigidity of finite covers of degree d of the M-dimensional projective space of index 1, where d ≥ 5 and M ≥ 10, that have at most quadratic singularities of rank ≥ 7 and satisfy certain regularity conditions. Up to now, only cyclic covers have been studied in this respect. The set of varieties that have worse...

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no rational maps $X\dashrightarrow V$ of degree 2, where $X$ is a rationally connected variety. This fact is true for many other families of primitive Fano varieties, either. It generalizes easily for rationally connected Galois r...

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy...

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dim...

In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$ and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant 7$, satisfying certain regularity conditions. Up to now, only cyclic covers were studied in this respect. The set of vari...

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dim...

We prove that a Fano complete intersection of codimension $k$ and index 1 in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement to the set of birationally superrigid complete intersections in the natur...

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to one, if $M\geqslant 2k+3$ and the maximum of the degrees of defining equations is at least 8. This is an essential improvements of the previous results about log canonical thresholds...

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of max...

We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity $(m-1)$, resolved by one blow up, is a restriction of some automorphism of the projective space ${\mathbb P}^M...

We sketch a proof of the following claim: every non-trivial structure of a
rationally connected fibre space (and so every structure of a Mori-Fano fibre
space) on a general (in the sense of Zariski topology) hypersurface of degree
$M$ in the $(M+1)$-dimensional projective space for $M\geq 14$ is given by a
pencil of hyperplane sections. In particul...

In this paper the codimension of the complement to the set of factorial hypersurfaces of degree $d$ in ${\mathbb P}^N$ is estimated for $d\geqslant 4$, $N\geqslant 7$.

The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2\mu$, where $\mu$ is the multiplicity of the singular point.

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 $\frac12 (M-9)(M-10)-1$.

We develop the quadratic technique of proving birational rigidity of
Fano-Mori fibre spaces over a higher-dimensional base. As an application, we
prove birational rigidity of generic fibrations into Fano double spaces of
dimension $M\geqslant 4$ and index one over a rationally connected base of
dimension at most $\frac12 (M-2)(M-1)$. An estimate fo...

We state and consider the Gabrielov–Khovanskii problem of estimating the multiplicity of a common zero for a tuple of polynomials in a subvariety of a given codimension in the space of tuples of polynomials. For a bounded codimension we obtain estimates of the multiplicity of the common zero, which are close to optimal ones. We consider certain gen...

We show that the global log canonical threshold of generic Fano complete
intersections of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to
1 if $M\geqslant 3k+4$ and the highest degree of defining equations is at least
8. This improves the earlier result where the inequality $M\geqslant 4k+1$ was
required, so the class of Fano complet...

In this paper we prove birational rigidity of large classes of Fano-Mori
fibre spaces over a base of arbitrary dimension, bounded from above by a
constant that depends on the dimension of the fibre only. In order to do that,
we first show that if every fibre of a Fano-Mori fibre space satisfies certain
natural conditions, then every birational map...

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in
${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not
birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the
parameter space.

We obtain upper bounds for the multiplicity of an isolated solution of a
system of equations $f_1=...= f_M =0$ in $M$ variables, where the set of
polynomials $(f_1,..., f_M)$ is a tuple of general position in a subvariety of
a given codimension which does not exceed $M$, in the space of tuples of
polynomials. It is proved that for $M\to\infty$ that...

We prove birational superrigidity of generic Fano complete intersections $V$
of type $2^{k_1}\cdot 3^{k_2}$ in the projective space ${\mathbb
P}^{2k_1+3k_2}$, under the condition that $k_2\geq 2$ and $k_1+2k_2=\mathop{\rm
dim} V\geq 12$, and of a few families of Fano complete intersections of
dimension 10 and 11. This is the third version: minor co...

It is proven that any structure of a fibre space into varieties of Kodaira
dimension zero on a generic Fano complete intersection of index one and
dimension $M$ in ${\mathbb P}^{M+k}$ for $M\geq 2k+1$ is a pencil of hyperplane
sections. We also describe $K$-trivial structures on varieties with a pencil of
Fano complete intersections.

We prove that a generic (in the sense of Zariski topology) Fano complete
intersection $V$ of the type $(d_1,...,d_k)$ in ${\mathbb P}^{M+k}$, where
$d_1+...+d_k=M+k$, is birationally superrigid if $M\geq 7$, $M\geq k+3$ and
$\mathop{\rm max} \{d_i\}\geq 4$. In particular, on the variety $V$ there is
exactly one structure of a Mori fibre space (or a...

This paper gives a survey of the modern theory of birational rigidity for Fano fibre spaces over a base of positive dimension. It is a sequel to a previous survey on birational rigidity of Fano varieties. Here techniques of the method of maximal singularities are described for Fano fibre spaces.
Bibliography: 53 titles.

We consider the connection between the problem of estimating the multiplicity of an algebraic subvariety at a given singular
point and the problem of describing birational maps of rationally connected varieties. We describe the method of hypertangent
divisors which makes it possible to give bounds for the multiplicities of singular points. The conc...

For certain families of Fano complete intersections we prove an estimate from below for the global (log) canonical threshold,
which implies the existence of Kähler-Einstein metric on generic varieties in these families.
Key wordsKähler-Einstein metric-Fano variety-projective space-complete intersection-global canonical threshold-log canonical thre...

In this survey paper birational geometry of higher-dimensional rationally connected varieties is discussed. In higher dimensions
the classical rationality problem generalizes to the problem of description of the structures of a rationally connected fiber
space on a given variety. We discuss the key concept of birational rigidity and present example...

In this paper, a complete proof of the so-called 8n
2-inequality is given, a local inequality for the self-intersection of a movable linear system at an isolated center of a noncanonical
singularity.

We prove birational rigidity of fiber spaces …: V !P1, the fiber of which is a Fano cyclic cover of index 1, provided it is suciently twist- ed over the base. In particular, non-rationality of these varieties is shown and their group of birational automorphisms is computed. The proof is ob- tained by a combination of the classical quadratic techniq...

We study birational geometry of Fano varieties, realized as double covers $\sigma\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\frac12 K_V|$. The group...

We give a complete proof of the so called 8n 2 -inequality, a local inequality for the self-intersection of a movable linear system at an isolated centre of a non canonical singularity. The inequality was suggested and several times published by I.Cheltsov but some of his arguments are faulty. We explain the mistake and replace the faulty piece by...

We prove the divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.

The theory of birational rigidity of rationally connected varieties generalises the classical rationality problem. This paper gives a survey of the current state of this theory and traces its history from Noether's theorem and the Lüroth problem to the latest results on the birational superrigidity of higher-dimensional Fano varieties. The main com...

We prove divisorial canonicity of Fano double hypersurfaces of general position. Comment: 27 pages, LaTeX

We prove that a smooth Fano hypersurface $V=V_M\subset{\Bbb P}^M$, $M\geq 6$,
is birationally superrigid. In particular, it cannot be fibered into uniruled
varieties by a non-trivial rational map and each birational map onto a minimal
Fano variety of the same dimension is a biregular isomorphism. The proof is
based on the method of maximal singular...

We prove birational superrigidity of direct products $V=F_1\times...\times
F_K$ of primitive Fano varieties of the following two types: either
$F_i\subset{\mathbb P}^M$ is a general hypersurface of degree $M$, $M\geq 6$,
or $F_i\stackrel{\sigma}{\to}{\mathbb P}^M$ is a general double space of index
1, $M\geq 3$. In particular, each structure of a r...

It is proved that the groups of birational automorphisms are the same as
the groups of biregular automorphisms for two series of Fano varieties of dimension
4 and higher: double spaces of index 1 and double quadrics of index 1. From this it
follows that the varieties are not rational. The proof that the respective automorphism
groups are the same e...

In this paper it is proved that there is only one pencil of rational surfaces on a smooth three-dimensional variety fibred into Del Pezzo surfaces of degree 1, 2 or 3 by a morphism (Iskovskikh's conjecture), provided that the class of 1-cycles (where is the canonical class and is the class of a line in a fibre) is not effective for any . This condi...

Iterating the procedure of making a double cover over a given variety, we
construct large families of smooth higher-dimensional Fano varieties of index
1. These varieties can be realized as complete intersections in various
weighted projective spaces. A generic variety in these families is proved to be
birationally superrigid; in particular, it adm...

We consider methods of analysis of controlled dynamical systems based on integral representations: the transition of the system
from one admissible state to another is associated with some integral transformations of the configuration space. The main
focus is on transforms with respect to integration over the Euler characteristic, which make it pos...

Combining the connectedness principle of Shokurov and Kollár, we develop a new technique of studying birational maps of natural
Fano fiber spaces. We prove that the only structures of a rationally connected fibration on direct products of typical Fano
varieties are projections onto the factors. In particular, the groups of biregular and birational...

We complete the study of birational geometry of Fano fiber spaces $\pi\colon
V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index
1. For each family of these varieties we either prove birational rigidity or
produce explicitly non-trivial structures of Fano fiber spaces. A new linear
method of studying movable systems on Fa...

We prove birational superrigidity of generic Fano fiber spaces
V/\mathbb P1{V/\mathbb {P}^{1}} , the fibers of which are Fano complete intersections of index 1 and dimension M in
\mathbb PM+k{\mathbb {P}^{M+k}} , provided that M≥ 2k+1. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear...

We prove birational superrigidity of Fano cyclic covers of index 1 over hypersurfaces in the projective space.

The article develops the principles of optimal control theory for probability distributions on the configuration space of a controlled dynamical system. Necessary conditions for optimality are derived in the form of the Pontryagin maximum principle for various classes of problems. Analytical representations in the space of distributions and observa...

We continue to study birational geometry of Fano fibrations $\pi\colon V\to
{\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1.
For a majority of families of this type, which do not satisfy the condition of
sufficient twistedness over the base, we prove birational rigidity (in
particular, it means that there are no other st...

We give a brief survey of the concept of birational rigidity, from its origins in the two-dimensional birational geometry, to its current state. The main ingredients of the method of maximal singularities are discussed. The principal results of the theory of birational rigidity of higher-dimensional Fano varieties and fibrations are given and certa...

It is proved that a general Fano hypersurface V = V-M subset of P-M of index 1 with isolated singularities in general position is birationally rigid. Hence it cannot be fibred into uniruled varieties of smaller dimension by a rational map, and each Q-Fano variety V' with Picard number 1 birationally equivalent to V is in fact isomorphic to V. In pa...

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We develop foundations of the theory of discontinuous Hamiltonian systems appearing in the problems of optimal control. We consider analogs of the classical Poisson and Liouville theorems for discontinuous Hamiltonian systems. We study the local geometry of discontinuous dynamical systems and describe singularities in general position and the behav...

We prove that a fibration X \to \Bbb P_1, the general fiber of which is a
smooth Fano threefold, is rationally connected. The proof is based on a
generalization of Tsen's classical theorem: a fibration X/C over a curve the
general fiber of which is a Fano complete intersection in a product of weighted
projective spaces has a section.

It is proved that a general Fano hypersurface of index 1 (in the projective
space) with isolated singularities of general position is birationally rigid.
Therefore it cannot be fibered into uniruled varieties of a smaller dimension
by a rational map. The group of birational self-maps is either trivial or
cyclic of order two.

We prove that a generic (in the sense of Zariski topology) Fano complete intersection V of the type (d 1 ,⋯,d k ) in ℙ M+k , where d 1 +⋯+d k =M+k, is birationally superrigid if M≥7, M≥k+3 and max{d i }≥4. In particular, on the variety V there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of bira...

One of the main achievements of algebraic geometry over the last 30 years is the work of Mori and others extending minimal models and the Enriques-Kodaira classification to 3-folds. This book, first published in 2000, is an integrated suite of papers centred around applications of Mori theory to birational geometry. Four of the papers (those by Puk...

We prove that each fiber-wise birational correspondence between smooth fibrations into (Fano) hypersurfaces, which is biregular
on a generic fiber, is a fiber-wise isomorphism.

We prove an analogue of the Gauss-Ostrogradskii theorem for integration over the Euler characteristic, expressing the Euler characteristic of a manifold with boundary in terms of the zeros of a smooth dynamical system and its behaviour on the boundary. This result makes it possible to compute the Euler characteristic of a closed manifold via the be...

We prove the birational superrigidity of a general Fano fibration \pi\colon V\to\mathbf P^1 whose fibre is a Fano hypersurface W_M\subset\mathbf P^M of index 1. If the fibration is sufficiently twisted over the base \mathbf P^1, then V has no other structure of a fibration into rationally connected varieties. We also formulate and discuss conjectur...