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Del Pezzo surfaces of degree one and examples of Zariski multiples

Ichiro Shimada

Abstract

We construct examples of Zariski N-tuples with large N using the monodromy action of the Weyl group of type E8 on the set of 240 lines in a del Pezzo surface of degree one.

Del Pezzo surfaces of degree one and examples of Zariski multiples

Abstract

We construct examples of Zariski N-tuples with large N using the monodromy action of the Weyl group of type E8 on the set of 240 lines in a del Pezzo surface of degree one.

Paper Structure

This paper contains 22 sections, 16 theorems, 76 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Combinatorial type sigmak
  4. Main Theorem
  5. Del Pezzo surfaces
  6. Picard lattice
  7. Monodromy
  8. Family of cubic surfaces
  9. Family of quartic double planes
  10. Family of bi-anti-canonical models of del Pezzo surfaces of degree one
  11. Lines in a del Pezzo surface of degree one
  12. Bertini involution
  13. Orbit decomposition by the monodromy
  14. The action on $L(X_b)^{\{k\}}$
  15. The action on $\overline{L}(X_b)^{\{k\}}$
  16. ...and 7 more sections

Key Result

Theorem 1.1

For each integer $k$ satisfying $1<k<119$, there exists a Zariski $N(k)$-tuple $\mathord{\mathcal{Z}}_k$ consisting of plane curves of degree $7+2k$, where $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 4.1: Orbits in $\overline{L}(X_b)^{\{2\}}$
  • Figure 5.1: Birational map $\pi_p$

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 28 more