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Enriques surfaces with non-generic non-degeneracy

Riccardo Moschetti, Franco Rota, Luca Schaffler

TL;DR

This paper advances the understanding of Enriques surfaces by analyzing the non-degeneracy invariant nd(Y) and its refinements. It establishes that nd cannot increase under specialization in families and completes the calculation of nd for 155 $(\tau,\overline{\tau})$-generic Enriques surfaces, revealing 14 exceptional cases and the first instances of nd=9. By introducing Fnd and Mnd, the authors connect nd to Fano and Mukai polarizations, enabling upper bounds via contractions of rational curves and a refined lattice-theoretic approach. The work blends lattice theory, Borcherds-type chamber analysis, and K3 covers to bound nd in tau-tbar generic families and raises open questions about nd values 5 and 6 and their geometric realizability.

Abstract

We study the non-degeneracy invariant $\mathrm{nd}(Y)$ of complex Enriques surfaces in families. Our first main result shows that $\mathrm{nd}(Y)$ cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the $155$ families of $(τ,\overlineτ)$-generic surfaces introduced by Brandhorst and Shimada. Of the previously known $144$ cases, only $3$ satisfy $\mathrm{nd}(Y)\neq10$, which is the non-degeneracy invariant of a general Enriques surface. The remaining $11$ families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on $\mathrm{nd}(Y)$ by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with $\mathrm{nd}(Y)=9$.

Enriques surfaces with non-generic non-degeneracy

TL;DR

This paper advances the understanding of Enriques surfaces by analyzing the non-degeneracy invariant nd(Y) and its refinements. It establishes that nd cannot increase under specialization in families and completes the calculation of nd for 155 -generic Enriques surfaces, revealing 14 exceptional cases and the first instances of nd=9. By introducing Fnd and Mnd, the authors connect nd to Fano and Mukai polarizations, enabling upper bounds via contractions of rational curves and a refined lattice-theoretic approach. The work blends lattice theory, Borcherds-type chamber analysis, and K3 covers to bound nd in tau-tbar generic families and raises open questions about nd values 5 and 6 and their geometric realizability.

Abstract

We study the non-degeneracy invariant of complex Enriques surfaces in families. Our first main result shows that cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the families of -generic surfaces introduced by Brandhorst and Shimada. Of the previously known cases, only satisfy , which is the non-degeneracy invariant of a general Enriques surface. The remaining families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with .

Paper Structure

This paper contains 23 sections, 33 theorems, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The non-degeneracy invariant in families
  4. Computing the non-degeneracy invariant: Fano and Mukai models
  5. Enriques surfaces, non-degeneracy, Fano and Mukai polarizations
  6. Enriques surfaces and lattices
  7. Isotropic sequences on Enriques surfaces
  8. Fano and Mukai polarizations
  9. Fano polarizations
  10. Mukai polarizations
  11. A refinement of the non-degeneracy invariant
  12. Upper bounds for Fano and Mukai non-degeneracy invariants
  13. Enriques surfaces with finite automorphism groups: numerical Fano and Mukai polarizations
  14. Specialization of the Non-degeneracy invariant in families
  15. Specialization of tau,bartau-generic Enriques surfaces
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $Y\to T$ be a family of Enriques surfaces with $T$ an irreducible and Noetherian scheme, and let $0\in T$ be a closed point. Then, for a very general $t \in T$ we have that

Figures (3)

  • Figure 1: The $L_{10}$ lattice. The labeling of the roots is compatible with BS22.
  • Figure 2: An $E_8$ configuration.
  • Figure 3: A $D_8$ configuration.

Theorems & Definitions (81)

  • Theorem 1.1: Corollary \ref{['cor:family']}
  • Theorem 1.2: Theorem \ref{['thm:boundLatticePol']}
  • Theorem 1.3: Corollary \ref{['cor:ndBP']} and \ref{['cor:ndmlp1']}
  • Theorem 1.4: Theorem \ref{['thm:ndTauBarTau']}
  • Theorem 1.5: Theorem \ref{['thm:family']}
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2: DK25
  • Definition 2.3
  • Proposition 2.4
  • ...and 71 more