48 research outputs found
Algebraic Legendrian Varieties
Real Legendrian subvarieties are classical objects of differential geometry
and classical mechanics and they have been studied since antiquity. However,
complex Legendrian subvarieties are much more rigid and have more exceptional
properties. The most remarkable case is the Legendrian subvarieties of
projective space and prior to the author's research only few smooth examples of
these were known.
The first series of results of this thesis is related to the automorphism
group of any Legendrian subvariety in any projective contact manifold. The
connected component of this group (under suitable minor assumptions) is
completely determined by the sections of the distinguished line bundle on the
contact manifold vanishing on the Legendrian variety. Moreover its action
preserves the contact structure.
The second series of results is devoted to finding new examples of smooth
Legendrian subvarieties of projective space. The contribution of this thesis is
in three steps: First we find an example of a smooth toric surface. Next we
find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding.
Finally, we realise that both of these are special cases of a very general
construction: a general hyperplane section of a smooth Legendrian variety,
after a suitable projection, is a smooth Legendrian variety of smaller
dimension. By applying this result to known examples and decomposable
Legendrian varieties, we construct infinitely many new examples in every
dimension, with various Picard rank, canonical degree, Kodaira dimension and
other invariants.Comment: 116 pages, 6 figures. Author's PhD thesis (corrected and improved),
defended on Feb 7th, 2008. to appear in Dissertationnes Mathematica
Singular Curves of Low Degree and Multifiltrations from Osculating Spaces
In order to study projections of smooth curves, we introduce multifiltrations
obtained by combining flags of osculating spaces. We classify all
configurations of singularities occurring for a projection of a smooth curve
embedded by a complete linear system away from a projective linear space of
dimension at most two. In particular, we determine all configurations of
singularities of non-degenerate degree d rational curves in when
and . Along the way, we describe the Schubert cycles
giving rise to these projections.
We also reprove a special case of the Castelnuovo bound using these
multifiltrations: under the assumption , the arithmetic genus of any
nondegenerate degree curve in is at most .Comment: 34 pages, 11 tables, 2 figures; v2 added references and made minor
corrections; v3 more minor revisions, to appear in IMR