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The space of twisted cubics

Katharina Heinrich, Roy Skjelnes, Jan Stevens

Abstract

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space.

The space of twisted cubics

Abstract

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space.

Paper Structure

This paper contains 23 sections, 23 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Cohen-Macaulay curves
  3. The image of a finite map
  4. Fitting ideals
  5. CM-curves
  6. Functor of Cohen-Macaulay curves
  7. Tangent space to $\mathrm{CM}$
  8. Conductor
  9. Twisted cubics
  10. Tangent space calculations
  11. Plain double points
  12. The condition (R.C.)
  13. Singular sections of cubics
  14. Critical locus
  15. Singular cubics
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mathrm{CM}^{3t+1}_{\mathbb{P}^3}$ be the space of CM-curves in $\mathbb{P}^3$ (over an algebraically closed field of arbitrary characteristic) with Hilbert polynomial $3t+1$, and let $\mathrm{H} \subset \operatorname{Hilb}^{3t+1}_{\mathbb{P}^3}$ denote the twisted cubic component of the Hilber

Theorems & Definitions (58)

  • Theorem
  • Theorem 2.5: Hø nsen, Heinrich
  • Proposition 2.9
  • proof
  • Lemma 2.10
  • proof
  • Proposition 2.11
  • proof
  • Lemma 2.13
  • proof
  • ...and 48 more