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On Gizatullin's Problem for quartic surfaces of Picard rank $2$

Carolina Araujo, Daniela Paiva, Sokratis Zikas

TL;DR

This paper investigates when a nontrivial automorphism of a smooth quartic surface $S\subset\mathbb{P}^3$ of Picard rank $2$ lifts to a Cremona transformation of $\mathbb{P}^3$. It develops a volume-preserving version of the Sarkisov program for Calabi–Yau pairs and classifies the space curves on $S$ whose blowups can initiate Sarkisov links, yielding strong constraints on possible Cremona realizations. By analyzing the Picard lattice data, Pell-type equations, and explicit Sarkisov decompositions, the authors determine the automorphism groups for all quartics with $\rho(S)=2$ and provide a precise breakdown according to the discriminant $r$. The results show that Cremona realizations occur only in a narrow, explicitly described set of cases (finite for many $r$ and infinite in a few), offering a near-complete classification and a framework to understand the inertia and decomposition groups in this geometric context. The work connects the Gizatullin problem to concrete birational geometry constructions and highlights the rarity of Cremona-realizable automorphisms among quartic K3 surfaces with small Picard rank.

Abstract

In this paper we determine which automorphisms of general smooth quartic surfaces $S\subset \mathbb{P}^3$ of Picard rank $2$ are restrictions of Cremona transformations of $\mathbb{P}^3$.

On Gizatullin's Problem for quartic surfaces of Picard rank $2$

TL;DR

This paper investigates when a nontrivial automorphism of a smooth quartic surface of Picard rank lifts to a Cremona transformation of . It develops a volume-preserving version of the Sarkisov program for Calabi–Yau pairs and classifies the space curves on whose blowups can initiate Sarkisov links, yielding strong constraints on possible Cremona realizations. By analyzing the Picard lattice data, Pell-type equations, and explicit Sarkisov decompositions, the authors determine the automorphism groups for all quartics with and provide a precise breakdown according to the discriminant . The results show that Cremona realizations occur only in a narrow, explicitly described set of cases (finite for many and infinite in a few), offering a near-complete classification and a framework to understand the inertia and decomposition groups in this geometric context. The work connects the Gizatullin problem to concrete birational geometry constructions and highlights the rarity of Cremona-realizable automorphisms among quartic K3 surfaces with small Picard rank.

Abstract

In this paper we determine which automorphisms of general smooth quartic surfaces of Picard rank are restrictions of Cremona transformations of .

Paper Structure

This paper contains 12 sections, 33 theorems, 52 equations.

Table of Contents

  1. Notation & Conventions
  2. Acknowledgements
  3. K3 surfaces and their automorphisms
  4. The lattices of a K3 surface and the global Torelli Theorem
  5. Curves and Linear Systems on quartic surfaces
  6. Aut-general K3 surfaces
  7. K3 surfaces of Picard rank $2$
  8. Ingredients from Birational Geometry
  9. The Sarkisov Program
  10. Calabi-Yau pairs and volume preserving birational maps
  11. Sarkisov links centered on curves on quartic surfaces
  12. Gizatullin's problem for quartics with Picard rank $2$

Key Result

Theorem 2

Let $S\subset \mathbb{P}^3$ be a smooth quartic surface over $\mathbb{C}$ with Picard rank $2$ and discriminant $r$.

Theorems & Definitions (73)

  • Theorem 2
  • Proposition 3: =Proposition \ref{['prop:blancLamyCurves']}
  • Example 4: $r=41$
  • Example 5: $r=48$
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Morrison
  • Definition 1.4
  • Theorem 1.5: Gluing isometries
  • Theorem 1.6: Global Torelli Theorem
  • ...and 63 more