On the geometric genus of reducible surfaces and degenerations of surfaces to unions of planes

In this paper we study some properties of degenerations of surfaces whose general fibre is a smooth projective surface and whose central fibre is a reduced, connected surface $X \subset IP^r$, $r \geq 3$, which is assumed to be a union of smooth projective surfaces, in particular of planes. Our original motivation has been a series of papers of G. Zappa which appeared in the 1940-50's regarding degenerations of scrolls to unions of planes. Here, we present a first set of results on the subject; other aspects are still work in progress and will appear later. We first study the geometry and the combinatorics of a surface like $X$, considered as a reduced, connected surface on its own; then we focus on the case in which X is the central fibre of a degeneration of relative dimension two over the complex unit disk. In this case, we deduce some of the intrinsic and extrinsic invariants of the general fibre from the ones of its central fibre. In the particular case of $X$ a central fibre of a semistable degeneration, i.e. $X$ has only global normal crossing singularities and the total space of the degeneration is smooth, some of the above invariants can be also computed by topological methods (i.e., the Clemens-Schmid exact sequence). Our results are more general, not only because the computations are independent on the fact that $X$ is the central fibre of a degeneration, but also because the degeneration is not semistable in general.Comment: latex2e, 26 pages, 11 figure

Special scrolls whose base curve has general moduli

In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality.Comment: Latex2e, shorter versio

Umbral Moonshine and the Niemeier Lattices

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper, and in particular include the Mathieu moonshine observed by Eguchi-Ooguri-Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero.Comment: 181 pages including 95 pages of Appendices; journal version, minor typos corrected, Research in the Mathematical Sciences, 2014, vol.

Utilização de técnicas de segmentação e classificação de imagens LANDSAT-TM para o mapeamento do uso das terras na região de Campinas.

Os produtos do sensoriamento remoto tem sido, nas últimas décadas, a ferramenta mais utilizada e testada nos mapeamentos da cobertura terrestre. O desenvolvimento de novas máquinas e programas para auxiliar a interpretação de imagens de satélite, tem possibilitado a integração cada vez maior destas ferramentas. Este trabalho testou e discutiu o potencial e limitações do algoritmo de segmentação por crescimento de regiões, para o mapeamento da ocupação e uso das terras, em áreas heterogêneas e densamente ocupadas, utilizando imagens do satélite Landsat-TM, na escala 1:50.000. Através da segmentação de uma imagem Landsat-TM e da classificação supervisionada dos polígonos definidos pelo sistema, foi elaborado um mapa de uso e ocupação das terras, testado através da execução de levantamento de campo, em 88 dos 388 polígonos formados no processo. Este trabalho envolveu ainda uma etapa de tratamento numérico dos dados, visando quantificar a precisão do mapeamento efetuado e discutir os problemas encontrados nas rotinas de segmentação e classificação