The Gröbner fan of the Hilbert scheme

Abstract

We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gr öbner degenerations with respect to the given term order.Then, we introduce a polyhedral fan that we call Gröbner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper "Double-generic initial ideal and Hilbert scheme" [5] (Bertone, Cioffi, Roggero, Ann. Mat. Pura Appl. ( 4) 196(1), 2017) to give a lower bound on the number of irreducible components of the Hilbert scheme.

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“…, x n ], where in this appendix k may have any characteristic. Let B = B(n + 1) be the Borel subgroup of upper triangular matrices of G = GL(n + 1), those invertible linear maps sending x j → j i=1 α ij x i , where the α ij ∈ k. An ideal Galligo's theorem [21] that any ideal degenerates to a Borel ideal and Theorem C.1 are inspiration for approaches to the classification of Hilbert scheme components using Borel ideals, [8], [12,19,29], and recently [41,45].…”

“…0) degenerates to such an ideal, [21] or see [14,Sec.15.9]. Also called Borel-fixed ideals, they are a way to understand and classify components of the Hilbert scheme, [7,8,12,19,29,41,42,45]. They are the most degenerate of homogeneous ideals in polynomial rings k[x 1 , .…”