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Models of hypersurfaces and Bruhat-Tits buildings

Kletus Stern, Stefan Wewers

Abstract

We propose a new approach to constructing semistable integral models of hypersurfaces over a discrete non-archimedian field $K$. For each stable hypersurface over $K$ we define a stability function on the Bruhat-Tits building of ${\rm PGL}(K)$ and show that its global minima correspond to semistable hypersurface models over some extension of $K$. This extends work of Kollar and of Elsenhans and Stoll on minimal hypersurface models. In the case of plane curves and residue characteristic zero, our results give a practical algorithm for constructing a semistable model over a suitable extension field.

Models of hypersurfaces and Bruhat-Tits buildings

Abstract

We propose a new approach to constructing semistable integral models of hypersurfaces over a discrete non-archimedian field . For each stable hypersurface over we define a stability function on the Bruhat-Tits building of and show that its global minima correspond to semistable hypersurface models over some extension of . This extends work of Kollar and of Elsenhans and Stoll on minimal hypersurface models. In the case of plane curves and residue characteristic zero, our results give a practical algorithm for constructing a semistable model over a suitable extension field.
Paper Structure (28 sections, 18 theorems, 176 equations, 4 figures)

This paper contains 28 sections, 18 theorems, 176 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Models of hypersurfaces with respect to a discrete valuation
  3. Semistable hypersurface models
  4. The stability function
  5. Finding the right field extension
  6. Stability of hypersurfaces
  7. Definitions and conventions
  8. The numerical criterion
  9. Finding instabilities
  10. The spherical complex
  11. Semi-instabilities
  12. Valuations and buildings
  13. Spaces of valuations
  14. $\mathcal{B}_K$ as an affine building
  15. Extending the base field
  16. ...and 13 more sections

Key Result

Theorem 1.3

There exists a finite extension $L/K$ such that the base change of $X$, $X_L\subset\mathbb{P}^n_L$, has a semistable hypersurface model with respect to the unique extension of $v_K$ to $L$.

Figures (4)

  • Figure 2.1: The four cases of Proposition \ref{['prop:plane_curves']}. The colored area in any of the four triangles corresponds to nonzero coefficients of an example for $F$.
  • Figure 2.2: Finding a semistability in Example \ref{['exa:plane_curve']}.
  • Figure 2.3: The spherical apartment $A_\mathcal{E}$.
  • Figure 2.4: The three lines $L_0,L_1,L_2$ defined by two points on a conic.

Theorems & Definitions (47)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • proof
  • Proposition 1.5
  • Corollary 1.6
  • Corollary 1.7
  • proof
  • Theorem 1.8
  • Definition 2.1
  • ...and 37 more