47 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
Finite sets of -planes in affine space
Let be a subvariety of affine space whose irreducible
components are -dimensional linear or affine subspaces of .
Denote by the set of exponents of standard monomials
of . We show that the combinatorial object reflects the geometry of
in a very direct way. More precisely, we define a -plane in
as being a set , where
#J=d and for all . We call the -plane thus defined
to be parallel to . We show that the number of
-planes in equals the number of components of . This generalises a
classical result, the finiteness algorithm, which holds in the case . In
addition to that, we determine the number of all -planes in parallel
to , for all . Furthermore, we describe
in terms of the standard sets of the intersections
, where runs through .Comment: 31 pages, 8 figure
Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors
The Bialynicki-Birula cells on the Hilbert scheme H^n({A}^d) are smooth and
reduced in dimension d=2. We prove that there is a schematic structure in
higher dimension, the Bialynicki-Birula scheme, which is natural in the sense
that it represents a functor. Let \rho_i be the Hilbert-Chow morphism from the
Hilbert scheme H^n({A}^d) to Sym^n(A^1) associated with the i^{th} coordinate.
We prove that a Bialynicki-Birula scheme associated with an action of a torus T
is schematically included in the fiber over the origin if the
i^{th} weight of T is non positive. We prove that the monic functors
parametrizing families of ideals with a prescribed initial ideal are
representable.Comment: 18 pages, simplified proofs, noetherian assumptions removed,
bibliography improve
Connect Four and Graph Decomposition
We introduce the standard decomposition, a way of decomposing a labeled graph
into a sum of certain labeled subgraphs. We motivate this graph-theoretic
concept by relating it to Connect Four decompositions of standard sets. We
prove that all standard decompositions can be generated in polynomial time,
which implies that all Connect Four decompositions can be generated in
polynomial time