Borcherds' method for Enriques surfaces

enriques_surface_automorphism_group(SY2::ZZLat, SX::ZZLat; ample::Union{ZZMatrix,Nothing}=nothing)

Compute the automorphism group of an Enriques surface, its nef cone and its smooth rational curves.

Let $\pi: X \to Y$ be the universal cover of an Enriques surface and $\epsilon$ the Enriques involution. Let $S_Y$ be the numerical lattice of $Y$. This function computes the image of the natural map

\[\varphi_Y \colon \mathrm{Aut}_{s}(Y) \to O(S_Y \otimes \mathbb{F}_2)\]

where $\mathrm{Aut}_{s}(Y) \leq \mathrm{Aut}(Y)$ denotes the subgroup of semi-symplectic automorphisms. Note that the kernel of $\varphi_Y$ is finite.

See [BS22], [BS22*1], and [BRS23] for background and algorithms.

Input:

• SY2 – the invariant lattice of the Enriques involution in the numerical lattice SX of $X$.
• ample – optionally an ample class as a $1\times x 10$ matrix representing an ample class w.r.t. the basis of SY2; if not given, some arbitrary Weyl-chamber is picked.

EnriquesBorcherdsCtx

Holds data for computing invariants of families of Enriques surfaces with fixed root invariants, in particular the automorphism group of a general element.

The setting is the following: Let $Y$ be an Enriques surface over an algebraically closed field of characteristic not $2$ and $π : X → Y$ its universal covering K3 surface. We denote by $S_Y$ and $S_X$ the numerical lattices of $Y$ and $X$.

Most importantly the type stores a chain of lattices

\[S_Y(2) \subseteq S_X \subseteq L_{1,25}\]

where $L_{1,15}$ is an even unimodular lattice of signature $(1,25)$.

An easy way to construct examples is to call [`generic_enriques_surface`](@ref).
Here the generic 1-nodal Enriques surface:

jldoctest julia> genericenriquessurface(1) Enriques Borcherds context with det(SX) = 1024 with root invariant [(:A, 1)]

```

borcherds_method(Y::EnriquesBorcherdsCtx; max_nchambers=-1)

Compute $\mathrm{Aut}_{s}(Y)$ of the Enriques surface $Y$ using Borcherds method.

Here $\mathrm{Aut}_{s}(Y)$ denotes the group consisting of semi-symplectic automorphisms of $Y$. The quotient $\mathrm{Aut}(Y)/\mathrm{Aut}_{s}(Y)$ is a finite group and known to be cyclic in most cases [BG24].

Let $\pi \colon X \to Y$ be the K3 cover of $Y$.

Input:

• Y – represents an Enriques surface in terms of $\pi^*\colon S_Y(2) \hookrightarrow S_X$.
• max_nchambers – abort the computation after max_nchambers chambers have been computed; return the generators, chambers and curves computed so far. They may not generate the full automorhism group, and not cover all orbits of chambers or rational curves.

Output:

1. A matrix group, the image of $\mathrm{Aut}_{s}(Y) \to O(S_Y)$
2. A complete list of representatives of the $G:=G_Y^0$-congruence classes of $S_Y|S_X$-chambers.
3. A list of $(-2)$-vectors representing smooth rational curves on $Y$ such that any rational curve of $Y$ is in the same $\mathrm{Aut}(Y)$-orbit as at least one class in the list.

Examples

Let $Y$ be an Enriques surface with finite automorphism group of type $I$. Assume that it is very general so that the covering K3 surface $X$ has picard rank $19$ and no extra non-symplectic automorphisms. In the terminology of [BS22] it is an $(E_8+A_1,E_8+A_1)$-generic Enriques surface, namely the one of Number 172 in Table 1.1. Following Section 7.4 of loc. cit. we compute some invariants.

The group $\mathrm{Aut}(Y)$ is a dihedral group of order $8$ and its image $\mathrm{Aut}^*(Y)$ in $O(S_Y)$ is a group of order $4$. It is this group we compute.

julia> Y = generic_enriques_surface(172)
Enriques Borcherds context
  with det(SX) = 4
  with root invariant [(:A, 1), (:E, 8)]

julia> autY, chambersY, rational_curves_reprY =  borcherds_method(Y);

julia> order(autY)
4

The nef-cone of $Y$ is is the union of the $\mathrm{Aut}(Y)$-orbits of the chambers in chambersY. Since the automorphism group of this particular Enriques surface $Y$ is finite, the nef-cone is a finite rational polyhedral cone. In our case there is a single chamber $D$ and it fixed by $\mathrm{Aut}(Y)$. Hence $D$ is the nef cone of $Y$. It has $12$ walls and each of them is defined by a smooth rational curve.

julia> D = only(chambersY);

julia> length(aut(D))
4

julia> length(walls_defined_by_rational_curves(D))
12

julia> length(walls(D))
12

The surface has $12$ rational curves and $9$ elliptic fibrations.

julia> rational_curvesY = gset(autY, (x,g) -> x*matrix(g), rational_curves_reprY);

julia> length(rational_curvesY)
12

julia> length(isotropic_rays(D))
9

The $\mathrm{Aut}(Y)$-orbits of the rational curves split as $2+2+2+2+4$.

julia> [length(i) for i in orbits(rational_curvesY)]
5-element Vector{Int64}:
 2
 2
 4
 2
 2

The elliptic fibrations split into $4$ orbits, as we shall see by two different methods:

julia> elliptic_classes = gset(autY,(x,g)->x*matrix(g), isotropic_rays(D));

julia> [length(i) for i in orbits(elliptic_classes)]
4-element Vector{Int64}:
 2
 1
 2
 4

julia> isomorphism_classes_elliptic_fibrations(Y)
4-element Vector{Tuple{Int64, Tuple{Vector{Tuple{Symbol, Int64}}, Vector{Tuple{Symbol, Int64}}}, TorQuadModuleElem}}:
 (2, ([(:E, 8)], []), [0 1 0 0 0 0 0 0 0 0])
 (135, ([(:A, 1)], [(:A, 7)]), [1 0 1 0 0 0 0 0 0 0])
 (270, ([(:D, 8)], []), [0 1 1 0 0 0 0 0 0 0])
 (120, ([(:E, 7)], [(:A, 1)]), [1 1 0 1 0 0 0 0 0 0])

Finally, we check the mass formula in this rather trivial case.

julia> mass(Y) == sum(1//length(aut(D)) for D in chambersY)
true

splitting_roots_mod2(Y::EnriquesBorcherdsCtx)

Return the image of the splitting roots of $Y$ in $S_Y \otimes \mathbb{F}_2$.

root_invariant(Y::EnriquesBorcherdsCtx)

Return the root invariant of $Y$.

mass(ECtx::EnriquesBorcherdsCtx)

Return the mass of $Y$

Let $V_0$ be a complete set of representatives of the $\mathrm{Aut}(Y)$-orbits of $L_{26}|S_Y$-chambers. The mass satisfies

\[\mathrm{mass}(Y)=\sum_{D \in V_0} \frac{1}{\sharp \mathrm{Aut}_G(D)} = \frac{\mathrm{ind}(D_0)}{\sharp \bar G_{X-}}\]

where $D_0$ is the initial chamber and $\mathrm{inv}(D_0)$ is its volume index.

rays(D::EnriquesChamber)

Return the list of primitive ray generators of $D$.

isotropic_rays(D::EnriquesChamber)

Return the list of primitive isotropic ray generators of $D$.

walls(D::EnriquesChamber)

Return the list of walls of $D$.

hom(D1::EnriquesChamber, D2::EnriquesChamber)

Return the set of elements of $G_Y^0$ mapping $D_1$ to $D_2$ where

\[G_Y^0 = \mathrm{Aut}^*(Y)\mathrm{W}(Y) = \{f \in O(S_Y) \mid f \mbox{ extends to an isometry of }S_X\mbox{ acting trivially on the discriminant group of }S_X\}\]

adjacent_chamber(D::EnriquesChamber, v::ZZMatrix) -> EnriquesChamber

Return return the chamber adjacent to D via the wall defined by v.

chamber_invariants(Y::EnriquesBorcherdsCtx)

Return a triple [volindex, name, root_type] of invariants of the induced chambers.

The invariants are taken from Table 1.1 and Table 1.2 of [BS22*1].

Output:

• The number of Vinberg chambers contained in $D_0$ is $O(S_Y \otimes \mathbb{F}_2)/\mathrm{volindex}$.
• The name of the embedding $S_Y(2) \to L_{1,25}$ stored in $Y$.
• The type of the root sublattice of the orthognal complement of $S_Y(2)$ in $L_{1,25}$.

isomorphism_classes_polarizations(Y::EnriquesBorcherdsCtx, h::ZZMatrix)

Return the isomorphism classes of numerical quasi-polarizations of $Y$ which are of the same type as $h$ along with some invariants.

See [BG24].

Output

The first return value is the degree of the the forgetful map

\[\mathcal{P}^h\colon \mathcal{M}_{En,h} \to \mathcal{M}_{En}, \quad (Y,h') \mapsto Y\]

of the moduli space of numerically $h$-quasi-polarized Enriques complex surfaces.

The second return value is a list of triples representing the fiber $(\mathcal{P}^h)^{-1}(Y)$

• the ramificaton index of $\mathcal{P}^h$ at $(Y,h')$
• the ADE types of the smooth rational curves orthogonal to $h'$
• a matrix $\bar g$ mod 2 such that $h' = h^{gw}$ for $w$ the unique element of the Weyl group of $Y$ such that $h^gw$ is nef.

Input:

• Y – representing the Enriques surface in question
• h – row matrix representing an element in $S_Y$ with respect to its basis matrix.

isomorphism_classes_elliptic_fibrations(Y::EnriquesBorcherdsCtx)

Return the list of isomorphism classes of elliptic fibrations on the Enriques surface $Y$. Let

\[\mathcal{P}^e \colon \mathcal{M}_{En,e} \to \mathcal{M}_{En}, \quad (Y,f) \mapsto Y\]

be the forgetful map from the moduli space of elliptic Enriques surfaces to the moduli space of Enriques surfaces. Here $f$ is the numerical class of a half-fiber.

Output:

A list of triples with the following entries:

• the ramification index of $\mathcal{P}^e$ in $(V,f)$
• the simple fibers, the double fibers
• representative of the half fiber $f$ modulo $2$

reducible_fibers(Y::EnriquesBorcherdsCtx, fbar::TorQuadModuleElem) -> simple fibers, multiple fibers

Return the ADE types and multiplicity of the reducible singular fibers of the genus-1-fibration induced by $f$.

Input:

• fbar – the class of $f/2$ in the discriminant group of $S_Y(2)$ where $f$ is the class of a half fiber of an elliptic fibration on $Y$

Output:

• The first return value is a list of the ADE-types of the simple fibers.
• The second return value is a list of the ADE-types of the double fibers.