526 research outputs found
Components of Gr\"obner strata in the Hilbert scheme of points
We fix the lexicographic order on the polynomial ring
over a ring . We define \Hi^{\prec\Delta}_{S/k},
the moduli space of reduced Gr\"obner bases with a given finite standard set
, and its open subscheme \Hi^{\prec\Delta,\et}_{S/k}, the moduli
space of families of #\Delta points whose attached ideal has the standard set
. We determine the number of irreducible and connected components of
the latter scheme; we show that it is equidimensional over ; and
we determine its relative dimension over . We show that analogous
statements do not hold for the scheme \Hi^{\prec\Delta}_{S/k}. Our results
prove a version of a conjecture by Bernd Sturmfels.Comment: 49 page
Schematic homotopy types and non-abelian Hodge theory
In this work we use Hodge theoretic methods to study homotopy types of
complex projective manifolds with arbitrary fundamental groups. The main tool
we use is the \textit{schematization functor} , introduced by the third author as a substitute for the
rationalization functor in homotopy theory in the case of non-simply connected
spaces. Our main result is the construction of a \textit{Hodge decomposition}
on . This Hodge decomposition is encoded in an
action of the discrete group on the object
and is shown to recover the usual Hodge
decomposition on cohomology, the Hodge filtration on the pro-algebraic
fundamental group as defined by C.Simpson, and in the simply connected case,
the Hodge decomposition on the complexified homotopy groups as defined by
J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity
property with respect to a weight filtration, generalizing the fact that the
higher homotopy groups of a simply connected projective manifold have natural
mixed Hodge structures. As a first application we construct a new family of
examples of homotopy types which are not realizable as complex projective
manifolds. Our second application is a formality theorem for the schematization
of a complex projective manifold. Finally, we present conditions on a complex
projective manifold under which the image of the Hurewitz morphism of
is a sub-Hodge structure.Comment: 57 pages. This new version has been globally reorganized and includes
additional results and applications. Minor correction
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