- Lunedì 29 Maggio, ore 11
- Martedì 30 Maggio, ore 11
- Giovedì 1 Giugno, ore 11
- Lunedì 5 Giugno, ore 11
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Abstract:
A classical tool to get numerical invariants of a curve singularity is the study of its value
semigroup. In case of a one branch singularity this semigroup is a numerical semigroup
(i.e. a submonoid of N with finite complement in it); in case the singularity has h branches,
this semigroup is a subsemigroup of Nh, belonging to the class of the so-called ”good semi-
groups”. Despite their name, the combinatoric of good semigroups is quite problematic;
moreover, for h ≥ 2, it is an open problem to understand which good semigroups can be
realized as value semigroups.
In case of a plane singularity with one branch, an old result of Ap ́ery shows that there
is a particularly strict connection between the value semigroups of the singularity and
of its blowup; this connection is obtained using a particular set of generators of the
semigroup, named ”Ap ́ery set”. In fact, using that result, it is possible to show very
easily, that the equisingularity classes given by the multiplicity sequence and by the
value semigroup coincide. In particular, this method allows to reconstruct the numerical
semigroup associated to a plane branch singularity starting from the multiplicity sequence.
When the singularity has more than one branch, in order to generalize the Ap ́ery result,
two main problems arise: firstly, the Ap ́ery set becomes an infinite set; secondly, in the
process of blowing-up it is necessary to deal with semilocal rings, that cannot be presented
as quotients of a power series ring in two variables, as it happens in the local case. These
problems where partially solved twenty years ago in the two branch case, but the general
case is still open.
In my talk, after describing some key definitions and results on value semigroups and
good semigroups of a curve singularity with h branches, I will explain explaining the
Ap ́ery process for a plane branch and then, I will present some recent results obtained in
a joint project with F. Delgado de la Mata, L. Guerrieri, N. Maugeri and V. Micale, that
are a significant progress toward a complete solution of this problem.
Dipartimento di Matematica e Informatica - University of Catania
V.le A. Doria 6, 95125
Catania
Italy
E-mail address: marco.danna@unict.it
Abstract. La superficie S di Cayley-Oguiso è stata introdotta come quartica determinantale da Cayley in alcune memorie pubblicate dal 1869 al 1871. S è stata poi reintrodotta in altro modo da Oguiso nel 2011 come esempio di superficie K3 con un un automorfismo g privo di punti fissi e con entropia positiva. In base a una congettura - dimostrata in alcuni casi - è possibile descrivere l'azione di un automorfismo simplettico o anti-simplettico di una K3 sul gruppo di Chow degli 0-cicli. In collaborazione con Robert Laterveer dimostriamo questa congettura nel caso di S e di g anti-simplettico.
]]>Birational geometry of symplectic varieties: smooth and singular versions
Abstract: In this talk, we will survey results on the birational geometry of symplectic varieties, that are hyperkähler manifolds in the smooth setting, and primitive symplectic varieties in the singular one. In particular, we will deal with the Kawamata-Morrison cone conjecture and the tools involved in its proof. The result in the singular setting is part of a joint work with Ch. Lehn and G. Pacienza.
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