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## On monodromy groups of del Pezzo surfaces

Cornell University
,
2017.

We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of a smooth surface are
hyperelliptic curves of genus $g\ge3$, then the monodromy group
acting on the integer $H^1$ on hyperplane sections is a proper
subgroup of $\mathrm{Sp}_{2g}(\mathbb Z)$.

Keywords: monodromyмонодромияdel Pezzo surfaceповерхность дель ПеццоГиперэллиптическая криваяj-invarianthyperelliptic curvej-инвариант

Publication based on the results of:

Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597-613

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...

Added: August 30, 2019

Trepalin A., Central European Journal of Mathematics 2014

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: October 14, 2013

Cheltsov I., Известия РАН. Серия математическая 2014 Т. 78 № 2 С. 167-224

We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities. ...

Added: December 6, 2013

Esterov A. I., Gusev G. G., Mathematische Annalen 2016 Vol. 365 No. 3 P. 1091-1110

We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued ...

Added: February 27, 2017

Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2011.

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated. ...

Added: October 11, 2013

Vyugin I. V., Левин Р. И., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 326-343

An analog of the classical Riemann-Hilbert problem formulated for classes of difference and q-difference systems is considered. We propose some strengthening of Birkhoff's existence theorem. ...

Added: August 18, 2017

Esterov A. I., Takeuchi K., Ando K., Advances in Mathematics 2015 Vol. 272 P. 1-19

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson. In particular, we compute the monodromy zeta-function. ...

Added: October 10, 2014

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31-48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

Takeuchi K., Esterov A. I., Lemahieu A., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1309.0630v4.

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain higher-dimensional analogues of their results, which, in particular, prove the conjecture for all isolated singularities of 4 variables, as well as for many classes of non-isolated and higher-dimensional singularities. One ...

Added: September 18, 2017

Brav C. I., Thomas H., Compositio Mathematica 2014 Vol. 150 No. 3 P. 343-333

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. ...

Added: September 29, 2014

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

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability. ...

Added: October 10, 2013

Lvovsky S., / Cornell University. Series math "arxiv.org". 2013. No. 1305.2205.

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. ...

Added: October 3, 2013

Gusein-Zade S., Journal of Algebra and its Applications 2018 Vol. 17 No. 10 P. 1-13

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring Bˆ(G) of a finite group G is defined. An ...

Added: October 27, 2020

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Andrey S. Trepalin, Central European Journal of Mathematics 2014 Vol. 12 No. 2 P. 229-239

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: December 3, 2013

Esterov A. I., Compositio Mathematica 2019 Vol. 155 No. 2 P. 229-245

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.
In particular, our result proves the multivariate ...

Added: February 5, 2019

Serge Lvovski, Manuscripta Mathematica 2014 Vol. 145 P. 235-242

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. ...

Added: October 14, 2014

Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313-326

We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...

Added: January 27, 2014

Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45-52

We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...

Added: September 26, 2019

Prokhorov Y., Annales de l'Institut Fourier 2015 No. 65 P. 1-16

We prove that for a Q-Gorenstein degeneration $X$ of del Pezzo surfaces, the number of non-Du Val singularities is at most $\rho(X)+2$. Degenerations with $\rho(X)+2$ and $\rho(X)+1$ non-Du Val points are investigated. ...

Added: October 17, 2014

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Algebraic Geometry 2014 Vol. 1 No. 1 P. 46-56

In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface ...

Added: October 10, 2013

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Vyugin I. V., Гонцов Р. Р., Успехи математических наук 2012 Т. 67 № 3 (405) С. 183-184

Получено обобщение результата Ильяшенко-Хованского, утверждающего, что разрешимость в квадратурах фуксовой системы с малыми коэффициентами эквивалентна ее треугольности. В работе этот результат обобщен на случай систем с малыми собственными значениями матриц вычетов. ...

Added: February 21, 2013

Zhgoon V., Федоров Г. В., Платонов В. П., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2018 Т. 482 № 2

We give a description of the cubic polynomials $f(x)$ with the coefficients in the quadratic number fields $\Bbb Q(\sqrt{5})$ and $\Bbb Q (\sqrt{-15})$, for which the continued fraction expansion of the irrationality $\sqrt{f(x)}$ is periodic. ...

Added: August 20, 2018