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Crepant resolutions of Weierstrass threefolds and non-Kodaira fibres

Andrea Cattaneo

TL;DR

The paper analyzes non-Kodaira fibres in smooth, equidimensional elliptic threefolds that resolve their Weierstrass models via crepant maps. It develops a framework based on rational Gorenstein and $cDV$ singularities and applies Reid–Mori crepant resolution theory to show that non-Kodaira fibres arise as contractions of Kodaira fibres, with their Kodaira type predicted by Tate's algorithm and located only over singular points of the reduced discriminant, $\Delta(\pi)_{\mathrm{red}}$. A central result is that crepant resolutions enforce $\Delta(\pi)=\Delta(p)$ and prevent certain divisorial contractions, tying fibre behavior tightly to the Weierstrass model. The work includes concrete examples illustrating these contractions, clarifying when they can and cannot occur, and connects to existing classifications in the literature, with implications for geometry and applications in areas like F-theory. Overall, it provides a precise partial classification of non-Kodaira fibres in this setting and demonstrates how crepant resolutions preserve key discriminant data while inducing predictable fibre contractions.

Abstract

In this paper we want to study the non-Kodaira fibres in a smooth equidimensional elliptic threefold. If the morphism to the Weierstrass model of the fibration is crepant, then we can locate the non-Kodaira fibres and give a description of their structure. In particular, they lie over the singular points of the (reduced) discriminant of the fibration and are contraction of a Kodaira fibre, whose type can be predicted using Tate's algorithm.

Crepant resolutions of Weierstrass threefolds and non-Kodaira fibres

TL;DR

The paper analyzes non-Kodaira fibres in smooth, equidimensional elliptic threefolds that resolve their Weierstrass models via crepant maps. It develops a framework based on rational Gorenstein and singularities and applies Reid–Mori crepant resolution theory to show that non-Kodaira fibres arise as contractions of Kodaira fibres, with their Kodaira type predicted by Tate's algorithm and located only over singular points of the reduced discriminant, . A central result is that crepant resolutions enforce and prevent certain divisorial contractions, tying fibre behavior tightly to the Weierstrass model. The work includes concrete examples illustrating these contractions, clarifying when they can and cannot occur, and connects to existing classifications in the literature, with implications for geometry and applications in areas like F-theory. Overall, it provides a precise partial classification of non-Kodaira fibres in this setting and demonstrates how crepant resolutions preserve key discriminant data while inducing predictable fibre contractions.

Abstract

In this paper we want to study the non-Kodaira fibres in a smooth equidimensional elliptic threefold. If the morphism to the Weierstrass model of the fibration is crepant, then we can locate the non-Kodaira fibres and give a description of their structure. In particular, they lie over the singular points of the (reduced) discriminant of the fibration and are contraction of a Kodaira fibre, whose type can be predicted using Tate's algorithm.

Paper Structure

This paper contains 8 sections, 17 theorems, 31 equations, 5 figures.

Table of Contents

  1. Gorenstein and canonical singularities
  2. Rational Gorenstein singularities
  3. Canonical singularities
  4. Crepant resolutions
  5. Crepant resolutions of threefolds with $cDV$ singularities
  6. Elliptic fibrations
  7. Crepant resolutions of Weierstrass threefolds
  8. Examples and non-examples

Key Result

Theorem 1

Let $\pi: X \longrightarrow B$ be a smooth equidimensional elliptic threefold, with morphism to the Weierstrass model $f: X \longrightarrow W$. If $f$ is a crepant resolution of $W$, then

Figures (5)

  • Figure 1: The singular fibres of the fibration (\ref{['equation: CCvG']}).
  • Figure 2: On the left a $I_0^*$ fibre, and on the right the non-Kodaira fibre of fibration (\ref{['equation: CCvG']}). The circled components are blown down: they are the multiplicity $1$ components which do not meet the section.
  • Figure 3: The singular fibres of the resolution of (\ref{['equation: weierstrass of fibration with I_1^*']}).
  • Figure 4: On the left a $I_1^*$ fibre, and on the right the non-Kodaira fibre on the resolution of (\ref{['equation: weierstrass of fibration with I_1^*']}). The circled components are contracted.
  • Figure 5: The fibre $F$.

Theorems & Definitions (58)

  • Theorem
  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4: cf. C3-f
  • Proposition 1.5: cf. C3-f
  • Example 1.6: Rational Gorenstein surface singularities
  • Example 1.7: Elliptic Gorenstein surface singularities
  • Definition 1.8
  • Proposition 1.9: cf. C3-f
  • ...and 48 more