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Conics on smooth quartic surfaces

Alex Degtyarev

TL;DR

The paper resolves a long-standing extremal problem for conics on smooth quartic surfaces by translating geometric configurations into square-4 vectors in 4-polarized Niemeier lattices and applying the global Torelli theorem. It introduces a refined lattice-theoretic framework (including the modified Néron–Severi lattice and geometric saturation) and develops sharp combinatorial bounds via block decompositions into $A_n$ and $D_n$ components, eliminating most Niemeier lattices and isolating the nine potential sources of large configurations. The main result is the exact bound $N_{4}(2)=800$, attained uniquely by the M$_{20}$-quartic, together with a classification of quartics with at least 720 conics and a detailed study of real conic counts, yielding the interval $656 eq N_{4}(2; ext{R}) eq 718$. The work combines deep lattice-theoretic methods (Nikulin discriminant forms, Vinberg’s algorithm) with substantial computer-aided enumeration (GAP) to realize a complete enumeration of extremal configurations and a near-complete map of the landscape of large conic configurations on quartics.

Abstract

We prove that the maximal number of conics, a priori irreducible of reducible, on a smooth spatial quartic surface is 800, realized by a unique quartic. We also classify quartics with many (at least 720) conics. The maximal number of real conics on a real quartics is between 656 and 718.

Conics on smooth quartic surfaces

TL;DR

The paper resolves a long-standing extremal problem for conics on smooth quartic surfaces by translating geometric configurations into square-4 vectors in 4-polarized Niemeier lattices and applying the global Torelli theorem. It introduces a refined lattice-theoretic framework (including the modified Néron–Severi lattice and geometric saturation) and develops sharp combinatorial bounds via block decompositions into and components, eliminating most Niemeier lattices and isolating the nine potential sources of large configurations. The main result is the exact bound , attained uniquely by the M-quartic, together with a classification of quartics with at least 720 conics and a detailed study of real conic counts, yielding the interval . The work combines deep lattice-theoretic methods (Nikulin discriminant forms, Vinberg’s algorithm) with substantial computer-aided enumeration (GAP) to realize a complete enumeration of extremal configurations and a near-complete map of the landscape of large conic configurations on quartics.

Abstract

We prove that the maximal number of conics, a priori irreducible of reducible, on a smooth spatial quartic surface is 800, realized by a unique quartic. We also classify quartics with many (at least 720) conics. The maximal number of real conics on a real quartics is between 656 and 718.
Paper Structure (35 sections, 8 theorems, 81 equations)

This paper contains 35 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Principal results
  3. Further observations and examples
  4. Idea of the proof
  5. Contents of the paper
  6. Acknowledgements
  7. Embedding to a Niemeier lattice
  8. Smooth quartics as $K3$-surfaces
  9. The modified Néron--Severi lattice
  10. Embedding $S\pdfstr{\sp\#}{^\sharp}$ to a Niemeier lattice
  11. Combinatorial bounds
  12. Niemeier lattices (see Conway.SloaneNiemeier)
  13. Orbits and combinatorial orbits
  14. The lattices $\mathbf{H}\sb n$, $\bA\sb n$, and $\bD\sb n$
  15. Counts and bounds blocks
  16. ...and 20 more sections

Key Result

lemma 1

The map $(S\ni h)\mapsto(S\ni h)\pdfstr{\sp\#}{^\sharp}$ is an involutive operation on the set of $h$-even $4$-polarized even lattices. A lattice $S$ is hyperbolic if and only if $(S\ni h)\pdfstr{\sp\#}{^\sharp}$ is positive definite.

Theorems & Definitions (14)

  • remark 1
  • lemma 1
  • lemma 2
  • proposition 1
  • proposition 2: see degt:conics
  • definition 1
  • definition 2
  • proposition 3
  • lemma 3
  • remark 2
  • ...and 4 more