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Almost minimal models of log surfaces

Karol Palka

Abstract

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and construct its almost minimal model, whose underlying surface has singularities not worse than $X$ and which differs from a minimal model by a contraction of some curves supported in the boundary only. For boundaries of type $rD$, where $D$ is reduced and $r\in [0,1]\cap \mathbb{Q}$, we show that if $X$ is smooth or $r\in [0,\frac{1}{2}]$ then the construction respects $(1-r)$-divisorial log terminality and $(1-r)$-log canonicity. We show that the assumptions are optimal, too.

Almost minimal models of log surfaces

Abstract

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface we define and construct its almost minimal model, whose underlying surface has singularities not worse than and which differs from a minimal model by a contraction of some curves supported in the boundary only. For boundaries of type , where is reduced and , we show that if is smooth or then the construction respects -divisorial log terminality and -log canonicity. We show that the assumptions are optimal, too.
Paper Structure (25 sections, 53 theorems, 57 equations, 15 figures)

This paper contains 25 sections, 53 theorems, 57 equations, 15 figures.

Table of Contents

  1. Main results
  2. Preliminaries
  3. Log Minimal Model Program for surfaces
  4. Coefficients and discrepancies
  5. Curves of the second kind
  6. Almost minimalization
  7. Relative MMP and reordering contractions
  8. Relative minimalization
  9. The definition of an almost minimalization
  10. Peeling and squeezing
  11. Effective almost minimalization
  12. Analogues for curves of the second kind
  13. Almost minimalization for reduced boundaries
  14. Discriminants and log canonical subdivisors
  15. Comparing graphs of log terminal singularities
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(X,D)$ be a log surface with a reduced boundary and let $r\in [0,1]\cap \mathbb{Q}$. Assume that $(X,rD)$ is $(1-r)$-lc and that $X$ is smooth or $r\leqslant \frac{1}{2}$. Then the following hold.

Figures (15)

  • Figure 1: The boundary $D$ in Example \ref{['ex:partially_almost_minimal']}.
  • Figure 2: Degenerate rational cycles.
  • Figure 3: Degenerate segments. Thick lines indicate their components.
  • Figure 4: Log canonical subdivisors.
  • Figure 5: Proposition \ref{['prop:ful_description_redundant']}, cases (3), (4), (6). Thick line indicates $R=D-E-\mathscr{l}$, dashed line is $\mathscr{l}$.
  • ...and 10 more figures

Theorems & Definitions (111)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: Fujino-MMP_for_algebraic_log_surfaces
  • Lemma 2.2: Fujino-MMP_for_algebraic_log_surfaces
  • Definition 2.3: A log exceptional curve
  • Remark 2.4: Direct images of $\mathbb{Q}$-Cartier divisors
  • Definition 2.5: A run of a Minimal Model Program
  • Remark 2.6: Projectivity
  • Definition 2.7: $\varepsilon$-lc surfaces
  • Remark 2.8
  • ...and 101 more