On the postulation of s^d fat points in P^d

In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in the projective plane and proved it when r is a square. Iarrobino formulated a similar conjecture in any projective space P^d. We prove Iarrobino's conjecture when r is a d-th power. As a corollary, we obtain new counter-examples modeled on those by Nagata.Comment: 14 page

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 $\rho_i^{-1}(0)$ 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