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Finite generation of higher rank quasi-monomial valuations via the extended Rees algebra

Zhiyuan Chen

Abstract

In the algebraic theory of K-stability, one of the most challenging problems is to show the graded algebra associated with certain higher rank quasi-monomial valuations are finitely generated. In the global case of Fano varieties and local case of klt singularities, the finite generation has been proved for quasi-monomial valuations on models of qdlt Fano type. In this paper, we generalize these results using a different argument, by studying the extended Rees algebra via a more algebraic approach. As consequences, our results apply to fibrations of Fano type with singularities worse than qdlt, and graded algebras coming from the multi-section ring of arbitrary divisors.

Finite generation of higher rank quasi-monomial valuations via the extended Rees algebra

Abstract

In the algebraic theory of K-stability, one of the most challenging problems is to show the graded algebra associated with certain higher rank quasi-monomial valuations are finitely generated. In the global case of Fano varieties and local case of klt singularities, the finite generation has been proved for quasi-monomial valuations on models of qdlt Fano type. In this paper, we generalize these results using a different argument, by studying the extended Rees algebra via a more algebraic approach. As consequences, our results apply to fibrations of Fano type with singularities worse than qdlt, and graded algebras coming from the multi-section ring of arbitrary divisors.

Paper Structure

This paper contains 14 sections, 18 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. The finite generation problem
  3. The extended Rees algebra
  4. The structure of the graded ring
  5. Preliminaries
  6. Birational geometry
  7. Filtrations and extended Rees modules
  8. Valuations
  9. Simple-toroidal lc centers
  10. Main theorems
  11. Auxiliary birational models
  12. The flatness of extended Rees modules
  13. The graded algebra
  14. An example

Key Result

Theorem 1.1

Let $\Bbbk$ be a field of characteristic $0$, and $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme essentially of finite type over $\Bbbk$. Let $(Y, D)$ be an lc pair, and $g \colon Y \to S$ be a proper morphism such that $-(K_Y + D)$ is $g$-ample. Suppose $E_1, \ldots, E_r$ are reduced divisors be the multi-section ring. Assume that $Z = \bigcap_{i=1}^{r} E_i$ is irreducible with the generic

Theorems & Definitions (46)

  • Theorem 1.1: Theorem \ref{['finite_generation_thm']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['the_central_fiber_of_Rees_alg']}
  • proof : Sketch of the proof of Theorem \ref{['main_thm_2']}
  • Corollary 1.4: Corollary \ref{['graded_ring_from_polyhedral_divisorial_sheaf']}
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 36 more