arXiv subjects

Serge Lvovski

Publications and source records attributed to Serge Lvovski.

11 recordsLinked to original sources

arXiv2026

On holomorphic submersions with biholomorphic fibers

We give explicit examples of holomorphic submersions $p\colon Y\to X$ such that $Y$ and $X$ are complex manifolds, all the fibers of $p$ are biholomorphic to the unit disc in the complex plane, but the mapping $p$ is not holomorphically locally trivial. In our examples the complex manifolds $Y$ are neighborhoods of the zero section of a line bundle $L$ on $X$; the line bundle $L$ is not, in general, supposed to be positive or negative.

math.CV↗

arXiv2025

Stabilization of fields of meromorphic functions on neighborhoods of a rational curve

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve $C$ with positive self-intersection. We prove that there exists a neighborhood $U\supset C$ such that any meromorphic function defined on a connected neighborhood of $C$ in $U$ can be extended to a meromorphic function on the entire $U$.

math.CV↗

arXiv2025

On fields of meromorphic functions on neighborhoods of rational curves

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the field of meromorphic functions on $F$ is isomorphic to the field of rational functions in one or two variables over $\mathbb C$.

math.CV↗

arXiv2024

On algebraic and non-algebraic neighborhoods of rational curves

We prove that for any $d>0$ there exists an embedding of the Riemann sphere $\mathbb P^1$ in a smooth complex surface, with self-intersection $d$, such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb C$. We give two different constructions of such neighborhoods, either as blowdowns of a neighborhood of the smooth plane conic, or as ramified coverings of a neighborhood of a hyperplane section of a surface of minimal degree. The proofs of non-algebraicity of these neighborhoods are based on a classification, up to isomorphism, of algebraic germs of embeddings of $\mathbb P^1$, which is also obtained in the paper.

math.AG↗

arXiv2022

On threefolds with the smallest nontrivial monodromy group

Using an adjunction-theoretic result due to A.J.Sommese together with a proposition from SGA7, we obtain a complete list of smooth threefolds for which the monodromy group acting on $H^2$ of its smooth hyperplane section is $\mathbb Z/2\mathbb Z$. The possibility of such a classification was announced by F.L.Zak in 1991.

math.AG↗

arXiv2022

On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.

math.AG↗

arXiv2022

On non-projective small resolutions

We construct a large class of projective threefolds with one node (aka non-degenerate quadratic singularity) such that their small resolutions are not projective.

math.AG↗

arXiv2020

On Legendrian curves in $\mathbb P^3$

We show that if a smooth projective curve $C\subset\mathbb P^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$ is unique up to a linear automorphism) and $C$ is linearly normal (i.e., not an isomorphic linear projection of a smooth curve $C'\subset\mathbb P^n$, $n>3$, where $C'$ does not lie in a hyperplane) then $C$ is a twisted cubic or a line.

math.AG↗

arXiv2018

On monodromy in families of elliptic curves over $\mathbb C$

We show that if we are given a smooth non-isotrivial family of elliptic curves over~$\mathbb C$ with a smooth base~$B$ for which the general fiber of the mapping $J\colon B\to\mathbb A^1$ (assigning $j$-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on $H^1(\cdot,\mathbb Z)$ of the fibers) coincides with $\mathrm{SL}(2,\mathbb Z)$; if the general fiber has $m\ge2$ connected components, then the monodromy group has index at most~$2m$ in $\mathrm{SL}(2,\mathbb Z)$. By contrast, in \emph{any} family of hyperelliptic curves of genus $g\ge3$, the monodromy group is strictly less than $\mathrm{Sp}(2g,\mathbb Z)$. Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

math.AG↗

arXiv2016

Some remarks on osculating self-dual varieties

Let us say that a curve $C\subset\mathbb P^3$ is osculating self-dual if it is projectively equivalent to the curve in the dual space $(\mathbb P^3)^*$ whose points are osculating planes to~$C$. Similarly, we say that a $k$-dimensional subvariety $X\subset\mathbb P^{2k+1}$ is osculating self-dual if its second osculating space at the general point is a hyperplane and $X$ is projectively equivalent to the variety in $(\mathbb P^{2k+1})^*$ whose points are second osculating spaces to $X$. In this note we show that for each $k\ge 1$ there exist many osculating self-dual $k$-dimensional subvarieties in $\mathbb P^{2k+1}$.

math.AG↗

arXiv2014

On projections of smooth and nodal plane curves

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $ν\colon \hat C\to C$ is its normalization, and $π\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection $\mathrm{pr}_p\circ ν\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\mathrm{deg}\, C=3$, one should assume in addition that $\degπ\ne4$). We prove that the morphism $π$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ ramified over $C^*$, where $X$ is a smooth surface. As a by-product, we prove the Chisini conjecture for mappings ramified over duals to general nodal curves of any degree $\ge3$ except for duals to smooth cubics; this strengthens one of Victor Kulikov's results.

math.AG↗