80 Open problems and technical gaps

Remark 80.1 (Orientation) This section records the known open problems and computational obstacles of the Coble moduli program. Each entry is a genuine gap between the results established elsewhere in this document and the fully rigorous, coordinate-explicit statements one would want for parity with the Enriques degree-$2$ program of (Alexeev et al. 2025). The problems are grouped by theme: lattice-theoretic verifications, the enumeration of isotropic orbits and cusps together with the reflection-group combinatorics, the dlt and KSBA stable models, and the foundational equations and computational checks. None of the statements below is claimed to be settled; several cross-reference partial results already in the text, and in each case the residue that remains open is stated explicitly.

80.1 Lattice-theoretic gaps

Remark 80.2 (Open problem: isometry class and genus of the Coble lattices)

The invariants $(r,a,\delta)$ of the geometric lattice $S_\Co \cong (11,11,1)_1$ (signature $(1,10)$, Gram matrix $\operatorname{diag}(2,-2,\dots,-2)$, equivalently $\generators{-2}\oplus E_{10}(2)$) and of the transcendental lattice $T_\Co = S_\Co^{\perp \lkt} \cong (11,11,1)_2$ (signature $(2,9)$) are recorded in [[coble-lattice-table|the Coble lattice table]], and both satisfy $q_{S_\Co} \cong q_{T_\Co} \cong (\ZZ/2\ZZ)^{11}$ with $q_{S_\Co} = -q_{T_\Co} \bmod 2\ZZ$. What is not yet formally established is the isometry-class verification and the genus decomposition: whether the genus of $T_\Co$ contains a unique isometry class. Concretely:

compute the Gram matrices of $S_\Co$ and $T_\Co$ and verify their $(r,a,\delta)$ invariants and genus cardinality via Nikulin’s classification (the $r > a$ check for a $2$-elementary lattice) (Nikulin 1980);
confirm that $r > a$ forces a unique class in the genus, so that the reduction of isotropic-orbit questions to the discriminant form $q_{T_\Co}$ (the Sterk-orbit theorem (Theorem 55.9)) is justified.

Remark 80.3 (Open problem: explicit primitive embedding matrices) The chain of primitive embeddings \[ T_\Co \injects \ten \injects \tdp \injects \lkt \] is used throughout (the sequence-of-embeddings lemma (Lemma 48.2), the primitive-embedding lemma (Lemma 48.1)), and Nikulin’s apparatus (Nikulin 1980 Prop. 1.14.4, 1.15.2) is the intended tool for the uniqueness of the primitive embedding $T_\Co \injects \ten$ and for the surjectivity of $\Orth(L) \to \Orth(T_\Co)$. What remains is to exhibit the embedding matrices in coordinate bases: derive the explicit primitive-embedding matrices realizing the chain above, and verify their primitivity directly rather than by invocation. The same coordinate deficit affects the derivation of the explicit equations for $C$ and $X$ (below): the invariants are listed, but the primitivity of the lattice embeddings lacks a rigorous derivation in terms of coordinate bases.

Conjecture 80.4 (Open problem: the horizontal folding involution and its eigenspaces) There is an orthogonal involution $\theta$ on the K3 lattice $\lkt \cong U^{3}\oplus E_8^{2}$ whose invariant and coinvariant sublattices are \[ \lkt^{\theta} \cong T_\Co, \qquad \lkt^{-\theta} \cong S_\Co, \] and whose action on roots induces the horizontal folding of the $(18,0,0)_1$ Coxeter diagram, as a functorial consequence of the lattice isometry (cf. the folded-root definition (Definition 51.3), the root-folding lemma (Lemma 51.5), and the invariant/coinvariant-lattice definition (Definition 40.2)). The involution should swap the polarization generators between sectors, $h_\En \leftrightarrow h_\Co$. Existence and the eigenlattice description are settled: the pair $(-\id_{S_\Co}, \id_{T_\Co})$ preserves the gluing graph of the Coble primitive embedding, so it extends to an isometry $\theta_\Co$ of $\lkt$ with the stated eigenlattices (the folding-involution proposition (Proposition 42.12)). The open technical content is to write down the $22\times 22$ matrix of $\theta_\Co$ on the standard basis of $U^{3}\oplus E_8^{2}$, which requires exhibiting the primitive embedding and the gluing anti-isometry in coordinates, and then to compare the polarization classes and the roots of the Coxeter diagram in that basis.

80.2 Isotropic orbits, cusps, and reflection groups

Question 80.5 (Open problem: enumeration of isotropic orbits by Sterk's technique) How many orbits of primitive isotropic vectors does $T_\Co$ have under $\Orth(T_\Co)$, $\Orth^{*}(T_\Co)$, and the arithmetic group $\Gamma_\Co$? Sterk’s technique (Sterk 1991) determines these by analyzing the orbits of the images (lifts) in the discriminant group $A_{T_\Co} \cong (\ZZ/2\ZZ)^{11}$ under $\Orth(q_{T_\Co})$: for a $2$-elementary lattice with $r > a$ the genus contains a unique class and $\Orth(T) \to \Orth(q_T)$ is surjective (Nikulin 1980), so a primitive isotropic vector $v$ with $\div(v)=d$ is determined up to $\Orth(T)$ by the tuple $(\div(v),\, \bar v \in A_T,\, v^2 = 0)$. The finite half of this is settled: $A_{T_\Co}$ has $528$ isotropic classes, on which $\Orth(q_{T_\Co})$ acts with two orbits, of sizes $1$ and $527$ (the Coble $Q$-fiber proposition (Proposition 41.8), the isotropic-class orbit theorem (Theorem 41.11)), and every primitive isotropic vector of $T_\Co$ lands in the nonzero one (the primitive-isotropic-class proposition (Proposition 41.12)). Under the degree-$2$ Enriques group the same $528$ classes split as $[1, 2, 120, 135, 270]$ (the finite-image orbit theorem (Theorem 42.9)). The open work is the passage from these finite decompositions back to the lattice:

lift the $\Orth(q_{T_\Co})$-orbits to $T_\Co$ and verify that exactly one $\Orth^{*}(T)$-orbit exists in divisibility $2$, the Eichler criterion being unavailable because $T_\Co$ contains no copy of $U$ (the Eichler criterion (Theorem 39.15) and the remark following it);
decide whether the four nonzero orbits of $\gent$-induced classes lift to four orbits of primitive isotropic vectors or to more, an integral parabolic stabilizer being permitted to have proper image in the finite one;
verify that the $\Orth(T)$-orbits coincide with the $\Gamma_\Co$-orbits, so that the Baily–Borel $0$-cusp is unique.

That every primitive isotropic $v \in T_\Co$ has $\div_{T_\Co}(v) = 2$ is already known (the divisibility lemma (Lemma 63.2)), which fixes the divisibility datum but not the orbit count.

Remark 80.6 (Open problem: explicit generators of $\Gamma_\Co$ and uniqueness of the 1-cusp) The arithmetic group governing the Coble locus is the stabilizer of the polarization $h_\Co$ inside $\Orth(\ten)$, cut further by the horizontal folding involution $\theta$, \[ \Gamma_\Co = \Stab_{\Orth(\ten)}(h_\Co) \intersect Z_{\Orth(\ten)}(\theta), \] where in the Enriques sector $h_\En = e + f\in U(2)$ is the degree-$2$ polarization vector, of square $4$ on the K3 side (the discriminant description of $\gent$ (Proposition 42.7)), and $h_\Co\in K_S^{\perp}\iscontainedin\Pic(S)$ is the degree-$2$ Coble polarization (Definition 30.10), of square $2$ downstairs with K3 pullback $\tilde h_\Co = f^{*}h_\Co$ of square $4$. The class $h_\Co$ is not the plane class $H$, whose K3 pullback $e_0 = f^{*}H$ has square $2$; the two must be kept apart, and it is $h_\Co$ that is the analogue of $h_\En$. A source-backed candidate for $\Gamma_\Co$, defined from the Enriques side by the Heegner line rather than by $\theta$, is $\Gamma_\Co^\En(\delta)$ from the induced Coble subgroup definition (Definition 42.6); identifying the two is part of the open work below. An explicit representation of $\Gamma_\Co$ by matrix generators is presently a stub. The open work is to:

compute the stabilizer/centralizer intersection in the Enriques (equivalently $\lkt$) lattice to produce a minimal set of matrix generators for $\Gamma_\Co$, as the intersection of the reflection group $W(T)$ with the centralizer $Z(\theta)$ and the stabilizer of the primitive vector $\tilde h_\Co$ ($\tilde h_\Co^2 = 4$). The involution $\theta$ itself exists as a lattice isometry without any coordinate description (the folding-involution proposition (Proposition 42.12)), and the image of the resulting group in $\Orth(q_{T_\Co})$ is already known (the finite-image orbit theorem (Theorem 42.9)); what a generating set adds is the lattice group above that finite image;
verify the uniqueness of the $1$-cusp for $\Gamma_\Co$. For the full orthogonal group this is settled: $\Orth^+(T_\Co)$ is transitive on primitive isotropic planes and $J^{\perp}/J\cong\latI_{0,7}(2)\cong A_1^{\oplus 7}$ (the unpolarized-cusp theorem (Theorem 85.27), with the split maximal hypothesis supplied by the split-maximal proposition (Proposition 41.4)). The residue is the subgroup statement, which does not follow from the full-group one.

This refines, at the level of orbits and generators, the $1$-cusp correspondence $(7,7,1)_0 \mapsto (8,6,0)_0$ established in the $1$-cusp correspondence lemma (Lemma 63.5).

Remark 80.7 (The maximal parabolic subdiagrams of $S_\Co$, and what they count) The Coxeter diagram $\Gamma_r$ of $S_\Co = (11,11,1)_1$ has $12$ roots, trivial automorphism group, and two maximal parabolic subdiagrams, $\wE_8(2)\wA_1$ and $\wB_9(2)$, both of even ordinary type (the Coble Picard parabolic theorem (Theorem 53.18)). By the parabolic/isotropic correspondence theorem (Theorem 53.15) these correspond to the two $\Orth^+(S_\Co)$-orbits of primitive isotropic vectors in $S_\Co$, and by the isotropic-trichotomy theorem (Theorem 35.14) both associated quotients $v^{\perp}/v$ have invariants $(9,9,1)$, so the two orbits are not separated by the invariants of the boundary lattice.

These orbits are elliptic fibrations of the K3 cover (the elliptic-fibration/isotropic theorem (Theorem 53.24)); they are not the $0$-cusps of the Coble period space, which are the orbits of primitive isotropic vectors in the transcendental lattice $T_\Co$ of signature $(2,9)$. Of the latter there is exactly one under the full orthogonal group (the unpolarized-cusp theorem (Theorem 85.27)), by the split maximal argument of the split-maximal proposition (Proposition 41.4). The two lattices share the invariant triple $(11,11,1)$ but not the signature, and a count taken in one is not a count in the other.

What remains open is the geometric identification: which of the two elliptic fibration classes of the K3 cover is distinguished by the Coble structure, and how the pair of maximal parabolic subdiagrams matches the boundary data of the degree-$2$ polarized moduli space.

Question 80.8 (Open problem: the reflection group of each Sterk fundamental domain) Sterk’s published fundamental domains for the five cusps of $\fentwo$ have $12, 10, 12, 11, 14$ walls (Sterk 1991), while Vinberg’s algorithm applied to the corresponding hyperbolic quotients $\eta_j^{\perp}/\eta_j$ returns ten walls in each case. The quotients for $j = 2,3,4,5$ are mutually isometric, so their full Weyl groups are conjugate and cannot account for four inequivalent published diagrams. The open work is to identify, for each $j$, the reflection subgroup $W_j\leq W(\eta_j^{\perp}/\eta_j)$ for which the published diagram bounds a fundamental domain, to exhibit that domain as a union of chambers of the full group, and to determine the index $[W(\eta_j^{\perp}/\eta_j) : W_j]$ as a covolume ratio. Cusp $2$, to which the Coble $0$-cusp is claimed to correspond, is the case in which the two computations agree. The data and the derivation are in [[sterk-root-counts-and-computed-chambers]].

Remark 80.9 (Open problem: hyperbolic-quotient derivation of the cusp correspondence) The cusp correspondence between the Coble cusps $(9,9,1)$, $(7,7,1)$ and their Enriques predecessors is established in the cusp-correspondence theorem (Theorem 63.1) (via the divisibility computations of the $w_1^\perp$ calculation (Lemma 63.4) and the $1$-cusp correspondence lemma (Lemma 63.5)). An independent hyperbolic-quotient derivation would place the correspondence on a self-contained lattice footing rather than resting on the alignment of invariants:

compute the isometry type of the hyperbolic quotient $e^{\perp}/e$ directly for a primitive isotropic $e \in T_\Co$;
use $\div_{T_\Co}(e)$ to pin down the unique $\Orth(T_\Co)$-orbit and verify that the quotient matches the Enriques signatures $(10,8,0)_1$ for $0$-cusps and $(8,6,0)_0$ for $1$-cusps.

The residue that is genuinely open is the orbit-uniqueness input: the proof of the divisibility lemma used there (Lemma 63.3) presently assumes a unique $\Gamma_\Co$-orbit of isotropic vectors in $T_\Co$, which is exactly the enumeration left open above.

80.2.1 General questions on hyperbolic reflection groups

The entries above ask for specific subdiagram and orbit computations in the Coble and Enriques lattices. The following are the corresponding questions asked of hyperbolic Coxeter diagrams in general; each is open, and each bears on how far the specific computations can be pushed.

Question 80.10 (Open problem: the cost of enumerating maximal parabolic subdiagrams) Given a Coxeter–Vinberg diagram $\Sigma$ on $n$ vertices, what is the cost of determining its maximal parabolic subdiagrams, equivalently the ideal vertices of the chamber (the ideal-vertices/parabolic corollary (Corollary 52.7))?

Enumerating the $2^{n}$ subsets is an upper bound, and the subdiagram-inheritance corollary (Corollary 50.15) prunes it: a subset that fails to be elliptic can be discarded together with every subset containing it, so the search runs over the order ideal of elliptic subsets. No lower bound is known, and no polynomial-time algorithm is known even for the decision problem asked of the whole diagram:

decide whether a given hyperbolic Coxeter diagram has finite covolume (the Coxeter-polytope volume theorem (Theorem 52.10)) without enumerating its maximal parabolic subdiagrams;
exhibit a family of diagrams on which the elliptic order ideal is of exponential size, or show that it is polynomial for the diagrams of finite-covolume groups.

The concrete instance in this monograph is the uniqueness of $\wB_7(2)$ as a maximal parabolic subdiagram of the ten-node diagram $G_{S_\Co}$, stated above; the general question is whether such uniqueness claims admit an argument short of the subdiagram search.

Conjecture 80.11 (Open problem: Galois invariance of the maximal parabolic count) Let $\Sigma$ be a Coxeter–Vinberg diagram whose Gram form is defined over a totally real number field $K$, and let $\sigma\in\gal(K/\QQ)$. The conjecture is that $\Sigma$ and its conjugate $\sigma(\Sigma)$ have the same number of maximal parabolic subdiagrams.

By the Galois-conjugate Gram-form remark (Remark 50.29) the signature of $\sigma(G)$ is not determined by that of $G$, so the conjugate diagram need not be hyperbolic at all and the statement requires the hypothesis that it is. The open work is to determine the action of $\gal(K/\QQ)$ on the set of maximal parabolic subdiagrams — whether it permutes them, and if so with what orbits. The diagrams of this monograph have Gram field $\QQ$ or $\QQ(\sqrt2)$, the latter arising from the $m_{ij} = 4$ bonds of the $B$-type diagrams, so the smallest instance is the nontrivial automorphism of $\QQ(\sqrt2)$ acting on the folded Sterk diagrams.

Question 80.12 (Open problem: growth of the maximal parabolic count with the rank) How does the number of maximal parabolic subdiagrams of a hyperbolic Coxeter diagram grow with its rank $n$?

Is there a bound, exponential or otherwise, valid for all diagrams of rank $n$, and does the finite-covolume condition of the Coxeter-polytope volume theorem (Theorem 52.10) improve it? The recorded data for this monograph is the count of vertices at infinity in the CoxIter runs on the five folded Sterk diagrams and the three $0$-cusp lattices, all of rank $9$ or $10$; a family of increasing rank is needed before a growth statement can be made.

Question 80.13 (Open problem: which finite-covolume reflection groups are arithmetic) Which finite-covolume hyperbolic Coxeter groups are arithmetic?

The datum available from the diagram is the Gram form over its base field $K$ (the Coxeter-base-field definition (Definition 50.25)) together with its Galois conjugates (the Galois-conjugate Gram-form remark (Remark 50.29)). The open work is to decide arithmeticity from that datum, and to determine how the property interacts with the maximal parabolic structure of the diagram. The reflection groups appearing in this monograph act on lattices defined over $\ZZ$, so they are arithmetic; the question governs whether the techniques used here extend to the non-crystallographic base-ring examples (Example 50.28).

Question 80.14 (Open problem: exact covolumes of hyperbolic Coxeter polytopes)

Compute $\vol(P)$ exactly for a hyperbolic Coxeter polytope $P$ given by its diagram (the covolume definition (Definition 52.9)).

In even dimension the volume is a rational multiple of the Euler characteristic and is computable from the diagram; in odd dimension the Euler characteristic vanishes and supplies no information, which is the case for every nine-dimensional chamber recorded in the Computations section. The open work is:

closed-form volume formulas for specific families, through Schläfli’s volume differential or through polylogarithmic values;
the relation between $\vol(P)$ and the maximal parabolic structure of $\Sigma$;
a covolume computation sharp enough to decide the index $[W : W_j]$ of a reflection subgroup, which is the quantity that would settle the discrepancy between Sterk’s published fundamental domains and the computed Vinberg chambers.

Question 80.15 (Open problem: theta series of indefinite lattices) For a positive-definite lattice the theta series $\theta_L(q) = \sum_{v\in L} q^{v^2/2}$ converges and is a modular form. For an indefinite lattice each level set $L[k]$ can be infinite (see the Lattice Theory section), the series does not converge, and a regularization is required before any modular statement can be made.

The open work is to define a regularized theta series for the hyperbolic lattices of this monograph, determine its transformation behaviour, and relate it to the lattice invariants — discriminant form, genus, and the reflection group — that the rest of this document computes.

80.3 The GIT and Looijenga models of the Coble moduli space

Question 80.16 (Open problem: is the Coble arrangement $\mch^{*}_{10A_1}$ empty?) The GIT compactification of the moduli of ten-nodal sextics is the Looijenga compactification of $\Gamma_{10A_1}\backslash(D(T_\Co) - \mch^{*}_{10A_1})$, for the arrangement \[ \mch^{*}_{10A_1} = \mch_\infty \intersect D(T_\Co) \] cut out by the divisibility-two roots of $\Lambda_1 = H^{\perp\lkt}$ (the GIT–Looijenga theorem (Theorem 47.11)). Whether that arrangement is empty decides the shape of the compactification:

if $\mch^{*}_{10A_1} = \varnothing$, the Looijenga construction returns the Baily–Borel compactification (the Looijenga-compactification definition (Definition 54.11)), so the GIT boundary of the ten-nodal sextics consists of the $0$-cusp point and the $1$-cusp curve and nothing else;
if $\mch^{*}_{10A_1} \neq \varnothing$, the GIT compactification carries a boundary divisor for each $\Gamma_{10A_1}$-orbit in the arrangement, and is a proper semitoroidal model strictly between Baily–Borel and toroidal.

The question is a lattice computation in $\Lambda_1$: is there a divisibility-two root of $\Lambda_1$ whose orthogonal hyperplane meets $D(T_\Co)$ in a proper hyperplane section? It is not answered by the divisibility lemma (Lemma 63.2), which computes divisibility inside $T_\Co$ and says nothing about divisibility inside the larger lattice $\Lambda_1$. One reduction is available: every root of the root lattice $L$ has divisibility $1$ in $\Lambda_1$ (Yu et al. 2025, sec. 4.2), so no root arising from a node of the sextic contributes, which is why $\mch^{*}_{10A_1}$ is contained in the larger arrangement $\mch_{10A_1}$ of the occult-period-map theorem (Theorem 47.10).

Question 80.17 (Open problem: the position of the three semifans)

The Coxeter, KSBA, and Looijenga semifans of the Coble period domain are determined by three unrelated inputs, and no one of them constrains another (the Looijenga-compactification definition (Definition 54.11) and the remark following it). Once $\mch^{*}_{10A_1}$ is computed, three questions remain:

is the resulting Looijenga semifan the Coxeter semifan of a reflection group, as it would be if the traces of $\mch^{*}_{10A_1}$ on a cusp boundary were the walls of a chamber?
does the Looijenga semifan refine the KSBA semifan, coarsen it, or is it incomparable?
which sextic degenerations lie over which cusp? Shah’s classification of the semistable sextics is explicit (Shah 1980), and the cusp correspondence of the cusp-correspondence theorem (Theorem 63.1) is known, so tracing a named degeneration such as a double cubic to the $0$-cusp $(9,9,1)_1$ or the $1$-cusp $(7,7,1)_0$ is a finite computation rather than a new theory.

80.4 dlt and KSBA stable models

Remark 80.18 (Open problem: Coble-specific dlt models and pot geometry)

The degree-$2$ Enriques program modernizes Morrison’s flowerpots (Morrison 1981) into dlt stable pairs, but the Coble-specific dlt models are not yet detailed. The open geometric content is:

Coble pot geometry. Define the explicit stable pair $(\mathcal V, \mathcal D)$ whose “Pot” component is a rational Coble surface, carrying a $\tfrac14(1,1)$ cyclic quotient singularity corresponding to the nodes of the rational sextic $C$. This is the local singularity package already conjectured for the stable quotient in the quarter-singularity conjecture (Conjecture 66.13), seen here on the dlt model (cf. the dlt-involution-pair definition (Definition 58.4), the divisor-model definition (Definition 58.6)).
Stalk assembly. Describe the transition of the stalk assembly and the integral-affine configuration as the Enriques surface log-collapses onto the discriminant divisor $\Delta$, where the K3 cover becomes nodal.

Remark 80.19 (Open problem: monodromy invariants and stable models $B(\lambda)$) Stable limits of Coble surfaces arise as $S_2$-quotients of nodal K3 surfaces, parameterized by a monodromy invariant $\ell \in \check{\mch}$ (surgery sizes) through the construction $B(\lambda)$ of (Alexeev et al. 2025). The open work is to:

determine the map from the Coble polarization to the discretization $\ell$ on the dual complex. The surgery sizes are the pairings $\ell_i = h\cdot\alpha_i$ against the roots $\alpha_i$ of the relevant Coxeter diagram, so the first step is to record which polarization class is meant and in which lattice the pairing is taken: the downstairs class $h_\Co\in K_S^{\perp}$ or its K3 pullback $\tilde h_\Co\in S_\Co$ (the Coble-polarization definition (Definition 30.10)), the roots living in the lattice of the chosen cusp. An orthogonal direct-sum presentation on its own does not make $\ell = 0$;
verify the slc stability of the resulting stable pair $(Z, \varepsilon C)$ for specific surgery vectors $\ell$.

This is the discrete-datum side of the KSBA program whose stability obligations are recorded in the quarter-singularity conjecture (Conjecture 66.13) and the surrounding KSBA discussion.

80.5 Foundational equations and computational verification

Remark 80.20 (Open problem: explicit equations for the Coble curve and its K3 cover) A Coble surface $S$ is the blowup of $\PP^2$ at the ten $A_1$ nodes of an irreducible rational plane sextic \[ C = \theset{ F(x,y,z) = 0 }, \qquad F(x,y,z) = \sum_{i+j+k=6} a_{ijk}\, x^i y^j z^k, \] subject to the nodal conditions $F(p_m) = \partial_x F(p_m) = \partial_y F(p_m) = \partial_z F(p_m) = 0$ at the ten special positions $p_1,\dots,p_{10} \in \PP^2$; the moduli space of such sextics is $9$-dimensional, and explicit models may be sought from the Steiner sextic or from index-$2$ Halphen pencils (cf. the Coble–Halphen blowdown lemma (Lemma 59.3), the ten-nodal-sextic lemma (Lemma 34.1)). The K3 cover is the double cover $X \xrightarrow{2:1} S$ of $\PP^2$ branched along $C$, with equation $w^2 = F(x,y,z)$ in $\PP(1,1,1,3)$, whose ten $A_1$ nodes lie above the positions $p_m$ (at $w = 0$, $[x:y:z] = p_m$). What is missing is a worked instance: derive an explicit equation $F(x,y,z) = 0$ for a rational sextic with ten nodes together with the corresponding cover $w^2 = F$, realizing $C$ as the image of $\PP^1$ under a degree-$6$ map $(s:t) \mapsto [f_0 : f_1 : f_2]$. Such an instance would anchor the coordinate derivations demanded by the lattice-embedding and involution problems above.

Remark 80.21 (Lattice and orbit anchors for the computations)

The problems above share a small set of concrete starting data, recorded here for reference:

Isotropic vectors. In $S_\Co$, primitive isotropic lines are represented by $v = e_0 \pm e_i$ (with $e_0^2 = 2$, $e_i^2 = -2$).
Polarization basis. The degree-$2$ polarization is $h = e + f\in U(2) = \sdp\iscontainedin \sen$, with $e\cdot f = 2$ and $h^2 = 4$ (the discriminant description of $\gent$ (Proposition 42.7)); it lies in the invariant lattice, not in $\ten$, and must be identified in a basis compatible with $\theta$.
Heegner line. The complement realizing $T_\Co$ inside $\ten$ is $\delta^{\perp}$ for $\delta = u - w$ in the unimodular $U$ summand, and this line is unique up to $\gent$ (the explicit Heegner-vector lemma (Lemma 42.2), the Heegner-line uniqueness theorem (Theorem 42.4)).
Discriminant forms. $q_{S_\Co} \colon (\mathbb{F}_2)^{11} \to \QQ/2\ZZ$ and $q_{T_\Co} \colon (\mathbb{F}_2)^{11} \to \QQ/2\ZZ$, with isometry of complements forcing $q_{S_\Co} = -q_{T_\Co} \bmod 2\ZZ$.
Sterk orbit lift. For the $2$-elementary $T_\Co$, an orbit is determined by the tuple $(\div(v),\, \bar v \in A_{T_\Co},\, v^2 = 0)$, reducing the search to $\mathbb{F}_2$-vector-space orbits under $\Orth(q_{T_\Co})$.
$\Gamma_\Co$ generators. Computed as the intersection of the reflection group $W(T)$ with the centralizer $Z(\theta)$ and the stabilizer of the primitive vector $\tilde h_\Co = f^{*}h_\Co$, of square $4$.