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Automorphisms and Hodge isometries of surfaces with K2=pg=1

  • Jorge Duque Franco
  • Alvaro Liendo
  • Pedro Montero
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Abstract

Let S be a minimal complex surface of general type with KS2=pg(S)=1. By a theorem of Catanese, the canonical model of S is a complete intersection of two sextics in P(1,2,2,3,3) whose equations admit a normal form, and the global Torelli theorem fails for these surfaces. We describe Aut(S) in terms of the coefficients of the normal form, prove that |Aut(S)|≤ 72, and classify the surfaces attaining the bound: two surfaces with singular canonical model and a one-parameter family Xr of smooth surfaces. For the surfaces Xr we compute a basis of the primitive cohomology consisting of eigenvectors of a subgroup Z/6Z × Z/6Z of Aut(Xr), given by residues of explicit polynomials, together with the intersection form in this basis. We then show that the natural map from Aut(Xr) to the group of Hodge isometries of the primitive lattice modulo ±1 is injective but has image of index at least 331 776, so that the Torelli principle for automorphisms fails for these surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{franco2026automorphisms,
  title = {{Automorphisms and Hodge isometries of surfaces with $K^2=p_g=1$}},
  author = {Jorge Duque Franco and Alvaro Liendo and Pedro Montero},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33364},
  eprint = {2610.33364},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33364},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Automorphisms and Hodge isometries of surfaces with K^2=p_g=1
AU  - Jorge Duque Franco
AU  - Alvaro Liendo
AU  - Pedro Montero
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.33364
UR  - https://arxiv.org/abs/2610.33364
AB  - Let S be a minimal complex surface of general type with K_S^2=p_g(S)=1. By a theorem of Catanese, the canonical model of S is a complete intersection of two sextics in P(1,2,2,3,3) whose equations admit a normal form, and the global Torelli theorem fails for these surfaces. We describe Aut(S) in terms of the coefficients of the normal form, prove that |Aut(S)|≤ 72, and classify the surfaces attaining the bound: two surfaces with singular canonical model and a one-parameter family X_r of smooth surfaces. For the surfaces X_r we compute a basis of the primitive cohomology consisting of eigenvectors of a subgroup Z/6Z × Z/6Z of Aut(X_r), given by residues of explicit polynomials, together with the intersection form in this basis. We then show that the natural map from Aut(X_r) to the group of Hodge isometries of the primitive lattice modulo ±1 is injective but has image of index at least 331 776, so that the Torelli principle for automorphisms fails for these surfaces.
ER  -