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Geometric transitions and integrable systems
We consider {\bf B}-model large duality for a new class of noncompact
Calabi-Yau spaces modeled on the neighborhood of a ruled surface in a
Calabi-Yau threefold. The closed string side of the transition is governed at
genus zero by an Hitchin integrable system on a genus Riemann surface
. The open string side is described by a holomorphic Chern-Simons
theory which reduces to a generalized matrix model in which the eigenvalues lie
on the compact Riemann surface . We show that the large planar
limit of the generalized matrix model is governed by the same Hitchin
system therefore proving genus zero large duality for this class of
transitions.Comment: 70 pages, 1 figure; version two: minor change
Trisecant Lemma for Non Equidimensional Varieties
The classic trisecant lemma states that if is an integral curve of
\PP^3 then the variety of trisecants has dimension one, unless the curve is
planar and has degree at least 3, in which case the variety of trisecants has
dimension 2. In this paper, our purpose is first to present another derivation
of this result and then to introduce a generalization to non-equidimensional
varities. For the sake of clarity, we shall reformulate our first problem as
follows. Let be an equidimensional variety (maybe singular and/or
reducible) of dimension , other than a linear space, embedded into \PP^r,
. The variety of trisecant lines of , say , has
dimension strictly less than , unless is included in a
dimensional linear space and has degree at least 3, in which case
. Then we inquire the more general case, where is
not required to be equidimensional. In that case, let be a possibly
singular variety of dimension , that may be neither irreducible nor
equidimensional, embedded into \PP^r, where , and a proper
subvariety of dimension . Consider now being a component of
maximal dimension of the closure of \{l \in \G(1,r) \vtl \exists p \in Y, q_1,
q_2 \in Z \backslash Y, q_1,q_2,p \in l\}. We show that has dimension
strictly less than , unless the union of lines in has dimension ,
in which case . In the latter case, if the dimension of the space
is stricly greater then , the union of lines in cannot cover the whole
space. This is the main result of our work. We also introduce some examples
showing than our bound is strict
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