The matroid stratification of the Hilbert scheme of points on $$\mathbb {P}^1$$

Silversmith, Rob

doi:10.1007/s00229-021-01280-z

Cited by 4 publications

(7 citation statements)

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“…If d h(d) < ∞ (as will always be true in this paper), then there are finitely many strata, and they are Zariski-locally closed. In general, there may be countably many strata; see [Sil21].…”

Section: The Tropical Ideal Of An Edge Of the Spinementioning

confidence: 99%

“…An additional complexity is given by the fact that the graph depends on the underlying field; the T -graph of Hilb 10 (A 2 ) differs for K = Q and K = R; see [HM12,Example 2.11] and [Sil21,Theorem 5.11].…”

Section: Introductionmentioning

confidence: 99%

“…This can be thought of as a generalization of the matroid stratification of the Grassmannian [GGMS87]. Very little is known about the tropical stratification; see [Sil21,FGG].…”

Section: Introductionmentioning

confidence: 99%

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The spine of the T-graph of the Hilbert scheme of points in the plane

Maclagan¹,

Silversmith²

2021

Preprint

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The torus T of projective space also acts on the Hilbert scheme of subschemes of projective space. The T -graph of the Hilbert scheme has vertices the fixed points of this action, and edges connecting pairs of fixed points in the closure of a one-dimensional orbit. In general this graph depends on the underlying field. We construct a subgraph, which we call the spine, of the T -graph of Hilb m (A 2 ) that is independent of the choice of infinite field. For certain edges in the spine we also give a description of the tropical ideal, in the sense of tropical scheme theory, of a general ideal in the edge. This gives a more refined understanding of these edges, and of the tropical stratification of the Hilbert scheme.

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Section: The Tropical Ideal Of An Edge Of the Spinementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The spine of the T-graph of the Hilbert scheme of points in the plane

Maclagan¹,

Silversmith²

2021

Preprint

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“…Our examples, however, were arrived at by different methods, and can be best understood in terms of the T-graph of the Hilbert scheme ; this is the graph whose vertices correspond to torus-fixed points of , and whose edges correspond to torus-invariant curves linking these fixed points. The T -graph was defined in [2] and studied further in [19, 32, 35], mostly in dimension . To produce our examples, we constructed edges of the T -graph corresponding to curves in whose general point is a smooth point with small tangent space dimension.…”

Section: Introductionmentioning

confidence: 99%

Small elementary components of Hilbert schemes of points

Satriano

Staal

2023

Forum of Mathematics, Sigma

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We answer an open problem posed by Iarrobino, Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert', in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276] $\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$ with dimension less than $(n-1)(d-1)$ ? For each $n\geq 4$ , we construct an infinite class of elementary components in $\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$ producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev, Power sums, Gorenstein algebras, and determinantal loci (Springer-Verlag, Berlin, 1999), 221–226] of the form with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$ .

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“…This result was used in [FGG] to study the space of all degree-2 zero-dimensional homogeneous tropical ideals in two variables, what can be thought of as the tropical Hilbert scheme of 2 points in trop(P 1 ). Higher-degree zero-dimensional tropical ideals have been studied in [Sil21] and [FGG] in the case of homogeneous ideals in two variables, where interesting connections to Schur polynomials were developed in the realizable case.…”

Section: Introductionmentioning

confidence: 99%

Paving tropical ideals

Anderson

Rincón

2022

J Algebr Comb

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Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I) + 1 -we call them paving tropical ideals. We show that paving tropical ideals of degree d + 1 are in bijection with Z n -invariant d-partitions of Z n . This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with Z n -invariant 2-partitions of quotient groups of the form Z n /L. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zerodimensional tropical ideals. Data Availability Statement. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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The matroid stratification of the Hilbert scheme of points on $$\mathbb {P}^1$$

Cited by 4 publications

References 10 publications

The spine of the T-graph of the Hilbert scheme of points in the plane

The spine of the T-graph of the Hilbert scheme of points in the plane

Small elementary components of Hilbert schemes of points

Paving tropical ideals

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