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Higher codimension nef and effective cycles on the Hilbert scheme of 3 points in projective 3-space

Gwyneth Moreland

Abstract

Nef and effective cones of divisors have been the subject of much study. In contrast, their higher codimension analogues are much harder to compute and few examples exist in the literature. In this paper we compute the nef cones in codimensions 2 & 3 and the effective cones in dimensions 2 & 3 for the Hilbert scheme of three points in $\mathbb{P}^3$. Our computation generalizes results of Ryan & Stathis and requires a careful analysis of the PGL orbits in the Hilbert scheme, as well as a new basis of the Chow ring inspired by Mallavibarrena and Sols.

Higher codimension nef and effective cycles on the Hilbert scheme of 3 points in projective 3-space

Abstract

Nef and effective cones of divisors have been the subject of much study. In contrast, their higher codimension analogues are much harder to compute and few examples exist in the literature. In this paper we compute the nef cones in codimensions 2 & 3 and the effective cones in dimensions 2 & 3 for the Hilbert scheme of three points in . Our computation generalizes results of Ryan & Stathis and requires a careful analysis of the PGL orbits in the Hilbert scheme, as well as a new basis of the Chow ring inspired by Mallavibarrena and Sols.
Paper Structure (21 sections, 24 theorems, 115 equations, 12 figures, 5 tables)

This paper contains 21 sections, 24 theorems, 115 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Hilbert schemes of points
  3. Groups of cycles and positive cones
  4. Two bases for the Chow groups of $\textup{Hilb}_3 {\mathbb P}^2$
  5. Techniques for computing the effective and nef cones
  6. The Chow ring of $\textup{Hilb}_3 {\mathbb P}^3$
  7. Basic structure
  8. Lifts and pushforwards of the MS basis and extending in (co)dimension 2
  9. Extending the MS basis in (co)dimension 3.
  10. Analyzing the seven orbits
  11. Planar fat points
  12. Collinear schemes supported at a single point
  13. Collinear and nonreduced schemes
  14. Schemes supported at a single point
  15. Collinear schemes
  16. ...and 6 more sections

Key Result

Proposition 2.1

For $\textup{Hilb}_3 {\mathbb P}^n$, the cycle map from the Chow groups to singular homology groups is an isomorphism. In particular, numerical equivalence and rational equivalence align for $\textup{Hilb}_3 {\mathbb P}^n$, and a cycle in $A_k(\textup{Hilb}_3 {\mathbb P}^n)$ is determined by its intersection numbers with a basis of $A_{3n-k}(\textup{Hilb}_3 {\mathbb P}^n)$.

Figures (12)

  • Figure 2.3: Intersection tables for $A_5(\textup{Hilb}_3 {\mathbb P}^2) \times A_1(\textup{Hilb}_3 {\mathbb P}^2)$ and $A_4(\textup{Hilb}_3 {\mathbb P}^2) \times A_2(\textup{Hilb}_3 {\mathbb P}^2)$ using the EL basis.
  • Figure 2.4: Pictures showing a general member of each of $H, F, \phi, \psi$. A solid point denotes a fixed point that must be included in the scheme; an $\times$ denotes a fixed point that is not necessarily in the support. A solid line denotes a fixed line; a dashed line indicates a line that can vary, and is used for our purposes to show a collinearity requirement.
  • Figure 2.5: General members of each of the basis elements of $A_4(\textup{Hilb}_3 {\mathbb P}^2), A_2 (\textup{Hilb}_3 {\mathbb P}^2)$.
  • Figure 2.7: General members of the basis elements of $A_3(\textup{Hilb}_3 {\mathbb P}^2)$
  • Figure 3.9: A picture of a general member of each of the lifts of the MS basis elements of codimension $1$. Solid lines denote fixed lines/planes, and dashed lines denote planes that can vary. In the picture for $\widetilde{F}$, the locus of schemes with a length two subscheme coplanar with a line $L$, the solid line in the dashed plane denotes that the plane containing the two points can vary, but that the plane must always contain the solid line.
  • ...and 7 more figures

Theorems & Definitions (59)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.9
  • proof
  • ...and 49 more