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Geometric Galois Theory and Monodromy

Aula Beltrami: Tuesday, September 26, 2017

Il giorno 26 settembre 2017 alle ore 11.00
in Aula Beltrami del Dipartimento di Matematica
dell'Universita' di Pavia si terra' un workshop dal titolo:

Geometric Galois Theory and Monodromy.

11-11.50 Kei MIURA (NIT, Ube College) Quasi-Galois points.

Abstract: We introduce the notion of ``quasi-Galois points''. This is a generalization of the Galois point.
A point P in projective plane is said to be quasi-Galois for a plane curve C if C admits a non-trivial birational transformation which preserves the fibers of the projection from P. It is remarkable that quasi Galois points play an important role for studying the automorphism group of C. In this talk, we show that the automorphism group of several (famous) curves is generated by the groups belonging to quasi-Galois points.

12-12.50 Francesco BASTIANELLI (Università di Bari) Irrationality issues for projective varieties.

Abstract: We will discuss various birational invariants extending the notion of gonality to projective varieties
of arbitrary dimension, and measuring the failure of a given variety to satisfy certain rationality properties,
as e.g. being uniruled, rationally connected, or rational.
In particular, we will describe these invariants for various classes of varieties and we will focus on recent results in the case of hypersurfaces.

15-15.50 Satoru FUKASAWA (Yamagata University) A birational embedding of an algebraic curve into a projective plane with two Galois points.

Abstract: Let C be a (reduced, irreducible) smooth projective curve over an algebraically closed field. We consider a rational map f from C to the projective plane, which is birational onto its image. For a point P
in the projective plane, if the function field extension induced by the projection from P is Galois, then P is called a Galois point for f(C). This notion was introduced by Yoshihara in 1996. There are not so many examples of plane curves with two Galois points. In this talk, a criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. As an application, several new examples of plane curves with two Galois points are described.

coffee break

16.30-17.20 Takeshi TAKAHASHI (Niigata University) Weak Galois Weierstrass points whose semigroups
are generated by two integers.

Abstract:
Let C be a nonsingular projective curve of genus > 1 over an algebraically closed field of characteristic 0.
For a point P in C, the Weierstrass semigroup H(P) is defined as the set of non-negative integers n for which there exists a rational function f on C such that the order of the pole of f at P is equal to n, and f is regular away from P. A point P in C is referred to as a weak Galois Weierstrass point if P is a Weierstrass point and there exists a Galois morphism from C to the projective line such that P is a total ramification point of the morphism. Recently, I and Prof. Komeda are investigating weak Galois Weierstrass points whose
Weierstrass semigroups are generated by two positive integers, as a generalization of studies on
Galois points for plane curves defined by Prof. Yoshihara. In my talk, I will introduce some of these results.

17.30-18.20. Francesco POLIZZI (Università della Calabria) Monodromy representations and surfaces with maximal Albanese dimension.

Abstract: We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of suitable monodromy representations of higher genus braid groups in the symmetric group.

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