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Ordinary $r$--tuples in cyclic quotient surface singularities

José I. Cogolludo-Agustín, Tamás Lászlo, Jorge Martín-Morales, András Némethi

Abstract

We describe those Weil divisors of cyclic quotient surface singularities which are (abstract) $r$--tuple curve singularities.

Ordinary $r$--tuples in cyclic quotient surface singularities

Abstract

We describe those Weil divisors of cyclic quotient surface singularities which are (abstract) --tuple curve singularities.

Paper Structure

This paper contains 14 sections, 4 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminary. The delta invariant of curves and $r$--tuples
  3. Ordinary $r$--tuples
  4. The guiding question
  5. The question and first examples.
  6. The generic linear section on minimal rational singularities
  7. Cyclic quotient singularities
  8. Notation
  9. A family of ordinary $r$--tuples embedded in a cyclic quotient singularity
  10. Cyclic quotient singularities. The new results.
  11. The 'canonical' embedding of the cyclic quotient singularities
  12. A new way to generate the functions $\{g_j\}_j$
  13. An analytic plumbing realization of $\widetilde{X}$ and the functions $g_j$ on it
  14. The embedded curves

Key Result

Theorem 1.1.1

Fix the minimal resolution of a cyclic quotient singularity $(X,o)$. Assume that on each $E_v$ we put $r_v$ different transversal discs in $\widetilde{X}$$(1\leq v\leq s)$. Then the projection into $(X,o)$ forms a $(\sum_v r_v)$--tuple whenever the following inequality holds for any $1\leq v_1\leq v Here $-k_v$ denotes the self-intersection of $E_v$ in $\widetilde{X}$, and $\mathrm{val}_v$ is the

Theorems & Definitions (12)

  • Theorem 1.1.1
  • Example 4.2.1
  • Proposition 5.2.1
  • proof
  • Lemma 5.2.2
  • proof
  • Example 5.2.3
  • Remark 5.2.4
  • Theorem 5.5.2
  • Example 5.5.4
  • ...and 2 more