137 research outputs found
Nagata type statements
Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution
naturally lead him to a tantalizing conjecture that remains widely open after
more than half a century of intense efforts. Using Nagata's theorem as starting
point, and the conjecture, with its multiple variations, as motivation, we
explore the important questions of finite generation for invariant rings, for
support semigroups of multigraded algebras, and for Mori cones of divisors on
blown up surfaces, and the rationality of Waldschimdt constants. Finally we
suggest a connection between the Mori cone of the Zariski-Riemann space and the
continuity of the Waldschmidt constant as a function on the space of
valuations.Comment: 45 pages. These notes correspond to the course of the same title
given by the first author in the workshop "Asymptotic invariants attached to
linear series" held in the Pedagogical University of Cracow from May 16 to
20, 201
Complex projective surfaces and infinite groups
The paper contains a general construction which produces new examples of non
simply-connected smooth projective surfaces. We analyze the resulting surfaces
and their fundamental groups. Many of these fundamental groups are expected to
be non-residually finite. Using the construction we also suggest a series of
potential counterexamples to the Shafarevich conjecture which claims that the
universal covering of smooth projective variety is holomorphically convex. The
examples are only potential since they depend on group theoretic questions,
which we formulate, but we do not know how to answer. At the end we formulate
an arithmetic version of the Shafarevich conjecture.Comment: 29 pages, some comments and examples added LaTeX 2.0
Poincar{\'e} series and linking of Legendrian knots
On a negatively curved surface, we show that the Poincar{\'e} series counting
geodesic arcs orthogonal to some pair of closed geodesic curves has a
meromorphic continuation to the whole complex plane. When both curves are
homologically trivial, we prove that the Poincar{\'e} series has an explicit
rational value at 0 interpreting it in terms of linking number of Legendrian
knots. In particular, for any pair of points on the surface, the lengths of all
geodesic arcs connecting the two points determine its genus, and, for any pair
of homologically trivial closed geodesics, the lengths of all geodesic arcs
orthogonal to both geodesics determine the linking number of the two geodesics.Comment: Minor modifications, 78
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