Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The isomorphism problem of projective schemes and related algorithmic problems

Takehiko Yasuda

Abstract

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

The isomorphism problem of projective schemes and related algorithmic problems

Abstract

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.

Paper Structure

This paper contains 20 sections, 25 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-decidability of the isomorphism problem of projective schemes
  4. Hilbert schemes
  5. The Hilbert scheme of a projective space
  6. The universal family
  7. The Hilbert scheme of a general projective scheme
  8. The action of $\mathrm{GL}_{r}$
  9. Projective equivalence
  10. Hom schemes and Iso schemes
  11. One-dimensional schemes
  12. Varieties with a big canonical sheaf or a big anti-canonical sheaf
  13. Computing intersection numbers
  14. Positivity of invertible sheaves
  15. Global generation
  16. ...and 5 more sections

Key Result

Proposition 3.2

The isomorphism problem of projective schemes is semi-decidable

Theorems & Definitions (62)

  • Remark 1.1
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4: cf. stillmancomputing
  • Remark 3.5
  • Proposition 4.1
  • Proposition 4.2
  • ...and 52 more