Abelian canonical covers of minimal rational surfaces and 𝔽1 up to degree $8$
Abstract
We classify abelian canonical covers of degree at most $8$ of smooth surfaces of minimal degree, and of ℙ2 and Hirzebruch surfaces not embedded as surfaces of minimal degree. Let π: X ⟶ W be an abelian canonical cover of a smooth surface of minimal degree W with group G and degree n. Degrees n ≤ 4 were classified earlier; we complete the picture for 5 ≤ n ≤ 8: - No such covers exist if n is prime and n > 3, nor if G=ℤ8. - For G=ℤ6, ℤ2 × ℤ2 × ℤ2 and ℤ2 × ℤ4 we give the complete list of covers. Only W =ℙ2, 𝔽0 and 𝔽1 arise. The list contains several families with unbounded pg, most of them new; X is smooth for general covers with G=ℤ2 × ℤ2 × ℤ2, but singular if G=ℤ6 and, with one exception, if G=ℤ2 × ℤ4. When W is ℙ2 or a Hirzebruch surface not embedded as a surface of minimal degree, we show that abelian canonical covers exist only for n = 2 or for G =ℤ2 × ℤ2, and we classify the latter completely: they form infinitely many families, most new, with unbounded pg and, most of them, with unbounded irregularity. The slopes K2/χ of all these X accumulate at $6$, at 8, and at 8α/(α+1) for every integer α≥ 2. These results rest on more general ones, valid for X normal, locally Gorenstein and of arbitrary dimension, and W smooth with pg(W) = 0: we determine which branch divisors of an abelian canonical cover must vanish, and, for canonical covers whose π*𝒪X splits as a sum of line bundles, we describe the 𝒪W-module structure of π*𝒪X, computing it completely when W is ℙ2 or a Hirzebruch surface.
Related
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BibTeX
@misc{dorothee2026abelian,
title = {{Abelian canonical covers of minimal rational surfaces and $\mathbb F_1$ up to degree $8$}},
author = {Alexandre Dorothée and Francisco Javier Gallego},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35724},
eprint = {2609.35724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35724},
}
RIS
TY - UNPB
TI - Abelian canonical covers of minimal rational surfaces and 𝔽_1 up to degree $8$
AU - Alexandre Dorothée
AU - Francisco Javier Gallego
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35724
UR - https://arxiv.org/abs/2609.35724
AB - We classify abelian canonical covers of degree at most $8$ of smooth surfaces of minimal degree, and of ℙ^2 and Hirzebruch surfaces not embedded as surfaces of minimal degree. Let π: X ⟶ W be an abelian canonical cover of a smooth surface of minimal degree W with group G and degree n. Degrees n ≤ 4 were classified earlier; we complete the picture for 5 ≤ n ≤ 8: - No such covers exist if n is prime and n > 3, nor if G=ℤ_8. - For G=ℤ_6, ℤ_2 × ℤ_2 × ℤ_2 and ℤ_2 × ℤ_4 we give the complete list of covers. Only W =ℙ^2, 𝔽_0 and 𝔽_1 arise. The list contains several families with unbounded p_g, most of them new; X is smooth for general covers with G=ℤ_2 × ℤ_2 × ℤ_2, but singular if G=ℤ_6 and, with one exception, if G=ℤ_2 × ℤ_4. When W is ℙ^2 or a Hirzebruch surface not embedded as a surface of minimal degree, we show that abelian canonical covers exist only for n = 2 or for G =ℤ_2 × ℤ_2, and we classify the latter completely: they form infinitely many families, most new, with unbounded p_g and, most of them, with unbounded irregularity. The slopes K^2/χ of all these X accumulate at $6$, at 8, and at 8α/(α+1) for every integer α≥ 2. These results rest on more general ones, valid for X normal, locally Gorenstein and of arbitrary dimension, and W smooth with p_g(W) = 0: we determine which branch divisors of an abelian canonical cover must vanish, and, for canonical covers whose π_*𝒪_X splits as a sum of line bundles, we describe the 𝒪_W-module structure of π_*𝒪_X, computing it completely when W is ℙ^2 or a Hirzebruch surface.
ER -