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Cartwright-Sturmfels Hilbert Schemes

Ritvik Ramkumar, Alessio Sammartano

Abstract

Let S be the Cox ring of a product of r projective spaces. In this paper, we study the Cartwright-Sturmfels Hilbert schemes of S, which are multigraded Hilbert schemes that only parametrize radical ideals. Our main result shows that these Hilbert schemes are always smooth and irreducible if the Picard rank r is at most 2. This result can be seen as a multigraded analogue of the famous theorems of Fogarty and Maclagan-Smith, where the Picard rank replaces the dimension of the ambient space.

Cartwright-Sturmfels Hilbert Schemes

Abstract

Let S be the Cox ring of a product of r projective spaces. In this paper, we study the Cartwright-Sturmfels Hilbert schemes of S, which are multigraded Hilbert schemes that only parametrize radical ideals. Our main result shows that these Hilbert schemes are always smooth and irreducible if the Picard rank r is at most 2. This result can be seen as a multigraded analogue of the famous theorems of Fogarty and Maclagan-Smith, where the Picard rank replaces the dimension of the ambient space.

Paper Structure

This paper contains 7 sections, 14 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Multigraded Hilbert schemes
  3. Bigraded Hilbert schemes over a product of Grassmannians
  4. Cartwright-Sturmfels ideals
  5. Tangent space
  6. Cutting process and proof of the main theorem
  7. Future directions

Key Result

Theorem 1.2

If $S$ is a standard $\mathbb{Z}^2$-graded polynomial ring, then every Cartwright-Sturmfels Hilbert scheme of $S$ is smooth and irreducible.

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof : Proof of \ref{['ThmMultigradedHilbertSchemeOverZ']}
  • Remark 2.6: Base change
  • Definition 3.3
  • ...and 33 more