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Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups

Sheldon Katz, David R. Morrison

TL;DR

The paper classifies simple flops on smooth threefolds by reducing to Gorenstein threefold singularities with irreducible small resolutions and showing there are exactly six primitive types, distinguished by Kollár's length invariant. It achieves this via invariant theory of Weyl groups associated with A_{n-1}, D_n, and E_n root systems, constructing explicit semi-universal deformations and simultaneous resolutions, and deriving generators for the Weyl-invariant rings. The authors develop a detailed framework connecting distinguished polynomials, partial resolutions, and root-system decompositions, and provide explicit computations (notably for E_6, E_7, E_8) to realize preferred versal forms and to prove the six-type classification. The work also yields explicit invariant bases for the exceptional E-types and presents an algorithmic approach to implement these calculations in symbolic software, with broader implications for understanding the geometry of small resolutions and simple flops in threefolds.

Abstract

We classify simple flops on smooth threefolds, or equivalently, Gorenstein threefold singularities with irreducible small resolution. There are only six families of such singularities, distinguished by Koll{á}r's {\em length} invariant. The method is to apply invariant theory to Pinkham's construction of small resolutions. As a by-product, generators of the ring of invariants are given for the standard action of the Weyl group of each of the irreducible root systems.

Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups

TL;DR

The paper classifies simple flops on smooth threefolds by reducing to Gorenstein threefold singularities with irreducible small resolutions and showing there are exactly six primitive types, distinguished by Kollár's length invariant. It achieves this via invariant theory of Weyl groups associated with A_{n-1}, D_n, and E_n root systems, constructing explicit semi-universal deformations and simultaneous resolutions, and deriving generators for the Weyl-invariant rings. The authors develop a detailed framework connecting distinguished polynomials, partial resolutions, and root-system decompositions, and provide explicit computations (notably for E_6, E_7, E_8) to realize preferred versal forms and to prove the six-type classification. The work also yields explicit invariant bases for the exceptional E-types and presents an algorithmic approach to implement these calculations in symbolic software, with broader implications for understanding the geometry of small resolutions and simple flops in threefolds.

Abstract

We classify simple flops on smooth threefolds, or equivalently, Gorenstein threefold singularities with irreducible small resolution. There are only six families of such singularities, distinguished by Koll{á}r's {\em length} invariant. The method is to apply invariant theory to Pinkham's construction of small resolutions. As a by-product, generators of the ring of invariants are given for the standard action of the Weyl group of each of the irreducible root systems.

Paper Structure

This paper contains 10 sections, 21 theorems, 122 equations, 2 figures, 1 table.

Table of Contents

  1. Reid's lemma, Pinkham's construction, Kollár's invariant, and the statement of the main theorem.
  2. Root systems, Weyl groups, and distinguished polynomials.
  3. Rational double points and simultaneous resolution.
  4. Simultaneous resolutions for $A_{n-1}$ and $D_n$.
  5. Simultaneous resolutions for $E_6$, $E_7$, and $E_8$.
  6. Singularities of the simultaneous partial resolutions.
  7. Relations among distinguished polynomials.
  8. Proof of the main theorem.
  9. The computation of preferred versal form in the $E_n$ cases.
  10. Restricted polynomials and the main computation.

Key Result

Corollary 1

The general hyperplane section of $X$ is uniquely determined by the length of the singular point $P$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (28)

  • Definition 1
  • Remark 1
  • Definition 2
  • Corollary 1
  • Definition 3
  • Remark 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 18 more