gistfile1.txt

## Example 1: the principal result

### Current pattern

The central comparison appears late, inside a `problem` block, under the title “Target theorem B,” followed by six desired compatibilities.

coble\_heegner\_research\_report

### Better pattern

> **Conjecture 1.1 (Coble–Heegner comparison).**  
> Fix a primitive root $\alpha\in T_{\mathrm{En}}$ with $\alpha^2=-2$, and let  
> $\Delta_\alpha\subset\mathcal F_{\mathrm{En},2}$ be the corresponding  
> Heegner component. Let  
> $\mathfrak M_{\mathrm{Co},2}^{\circ}$ be the open moduli stack defined in  
> Definition 2.1, and let  
> $\overline{\mathfrak M}_{\mathrm{Co},2}^{\mathrm{KSBA},\nu}$  
> denote the normalization of its KSBA closure.
> 
> Then the period morphism induces an isomorphism
> 
> 

$$\mathfrak M_{\mathrm{Co},2}^{\circ} \simeq \Delta_\alpha^\nu,$$

> 
> and extends to an isomorphism
> 
> 

$$\overline{\mathfrak M}_{\mathrm{Co},2}^{\mathrm{KSBA},\nu} \simeq \bigl(\overline{\Delta_\alpha}^{\,\Sigma_{\mathrm{En}}}\bigr)^\nu,$$

> 
> where the right-hand side is the normalized semitoroidal closure defined in  
> §6. Under this isomorphism, the arithmetic groups, universal ramification  
> divisors, cusp lattices, and induced semifans agree.

This gives the reader a single expected answer. Subsequent sections can prove pieces of it or state exact hypotheses under which it follows.

## Example 2: the projective quotient and its dimension

### Current pattern

The report fixes a point $p$, subtracts the centralizer dimension, and then asks for a quotient by the full centralizer, without first specifying the invariant locus on which the full centralizer acts.

coble\_heegner\_research\_report

### Better pattern

> Let $F=Y^\tau$, and for $p\in F$ let $U_p$ be the open subset of  
> $\mathbf P H^0(Y,\mathcal O_Y(4,4))^\tau$ consisting of curves having one  
> ordinary node at $p$ and no other singularities. Put
> 
> 

$$U=\bigcup_{p\in F}U_p,\qquad G=Z_{\operatorname{Aut}(Y)}(\tau),\qquad G_p=\operatorname{Stab}_G(p).$$

> 
> Then $G$ acts on $U$, and the natural morphism
> 
> 

$$[U/G]\longrightarrow\mathfrak M_{\mathrm{Co},2}^{\circ}$$

> 
> is the projective presentation morphism.
> 
> **Lemma.** Each $U_p$ is a nonempty open subset of a hyperplane in  
> $\mathbf P^{12}$. If the generic $G_p$\-stabilizer is finite, then
> 
> 

$$\dim[U_p/G_p]=11-\dim G_p^\circ=9.$$

> 
> **Conjecture.** The projective presentation morphism is an isomorphism.

The centralizer now has a stated reason for appearing, the action is well defined, and the dimension computation supports a precise conjecture rather than standing alone.

## Example 3: the reflection twist

### Current pattern

> “The formula follows by separating $S_{\mathrm{En}}$, $\mathbf Z\alpha$, and $\alpha^\perp_{T_{\mathrm{En}}}$.”

### Better pattern

> Over $\mathbf Q$ there is an orthogonal decomposition
> 
> 

$$L_{K3,\mathbf Q} = S_{\mathrm{En},\mathbf Q} \oplus \mathbf Q\alpha \oplus T_{\mathrm{Co},\mathbf Q}.$$

> 
> Relative to these summands, the actions of $I_{\mathrm{En}}$ and  
> $w_\alpha$ are
> 
> 

$$I_{\mathrm{En}}=(+1,-1,-1),\qquad w_\alpha=(+1,-1,+1).$$

> 
> Hence
> 
> 

$$I_{\mathrm{Co}}=w_\alpha I_{\mathrm{En}}=(+1,+1,-1),$$

> 
> so
> 
> 

$$L_{K3,\mathbf Q}^{I_{\mathrm{Co}}=1} = S_{\mathrm{En},\mathbf Q}\oplus\mathbf Q\alpha, \qquad L_{K3,\mathbf Q}^{I_{\mathrm{Co}}=-1} = T_{\mathrm{Co},\mathbf Q}.$$

> 
> It remains to identify the integral intersections. By Lemma 4.3,
> 
> 

$$L_{K3}\cap \bigl(S_{\mathrm{En},\mathbf Q}\oplus\mathbf Q\alpha\bigr) = S_{\mathrm{En}}\oplus\mathbf Z\alpha,$$

> 
> and $T_{\mathrm{Co}}$ is primitive in $L_{K3}$. The claimed integral  
> eigensublattices follow.

This rewrite makes the rational argument transparent and exposes the precise integral lemma that the original proof omitted.

## Example 4: finite orbit computations

### Current pattern

The report gives finite orbit sizes and repeatedly warns that they are not cusp classifications.

coble\_heegner\_research\_report

### Better pattern

> Reduction modulo $T_{\mathrm{Co}}$ gives
> 
> 

$$r_1: \Gamma_{\mathrm{Co},2}\backslash \operatorname{IGr}_1(T_{\mathrm{Co}}) \longrightarrow \bar\Gamma_{\mathrm{Co},2}\backslash \operatorname{IGr}_1(A_{T_{\mathrm{Co}}}).$$

> 
> Proposition A.2 computes the target: it has four elements represented by  
> $\bar e_A,\bar e_B,\bar e_C,\bar e_D$, listed in Table A.1 together with  
> their stabilizers.
> 
> Thus every integral $0$\-cusp has one of four finite reduction types.  
> To obtain an integral classification, it remains to determine the fibers of  
> $r_1$. This is reduced in Proposition 5.4 to computing the image of the  
> integral parabolic stabilizer
> 
> 

$$\operatorname{Stab}_{\Gamma_{\mathrm{Co},2}}(e) \longrightarrow \operatorname{Stab}_{\bar\Gamma_{\mathrm{Co},2}}(\bar e).$$

The computation has a defined output and a defined role. No standalone warning is required.

## Example 5: correcting obsolete Gram data

### Current pattern

The text records that a “previously used” value $2$ was wrong and that $4$ gives the correct rank.

coble\_heegner\_research\_report

### Better pattern

> **Lemma 6.2.** In the square-cusp lattice,
> 
> 

$$(\alpha_{20},\alpha_{21})=4.$$

> 
> **Proof.** Both roots have square $-4$, and the thick Coxeter edge between  
> them has label $m=\infty$; with the sign convention of §6.1 this gives  
> $(\alpha_{20},\alpha_{21})=4$. Equivalently, substituting this value in the  
> displayed Gram matrix gives corank $2$, as required for the rank-$18$  
> cusp lattice. The value $2$ gives corank $0$ and is therefore  
> incompatible with the ambient lattice.

If historical provenance matters, add one sentence in a changelog:

> This corrects the input matrix used in computations prior to version $v_3$.

That information does not belong in the main logical flow.

# III. Style guide for a revised mathematical report

## 1\. State the governing assertion before developing the machinery

The first section should contain the exact moduli objects, the principal conjecture or theorem, the strongest proved partial result, and a one-paragraph proof roadmap. Do not postpone the actual target until after the background theory.

A useful first-page pattern is:

> We study the normalization of the KSBA closure of the degree-$2$ Coble locus. The open locus admits two descriptions: as the normalization of a $(-2)$\-Heegner component in degree-$2$ Enriques moduli and as a moduli space of singular Coble pairs. Conjecture 1.1 identifies these descriptions and their semitoroidal compactifications. The principal results proved here are Propositions 1.2–1.4. The remaining obstruction is the integral cusp-lifting theorem of Conjecture 1.5.

The AEGS paper is a relevant local model: it identifies its moduli problem and compactification objective at the outset, then tells the reader how the period, integral-affine, and stable-pair constructions contribute. [![](https://www.google.com/s2/favicons?domain=https://arxiv.org&sz=128)arXiv](https://arxiv.org/abs/2312.03638)

## 2\. Define only what the intended audience needs defined

For standard objects, cite and fix conventions:

> We use the terminology of KSBA pairs and dlt models as in \[reference\], and the $2$\-elementary lattice conventions of \[Nikulin, §…\].

Define in full:

*   nonstandard moduli objects;

*   competing meanings of “Coble surface”;

*   the distinguished divisors and involutions;

*   unusual normalizations or sign conventions;

*   any object whose exact category affects the theorem.

Do not re-teach K3 surfaces, normality, $A_1$\-singularities, or Baily–Borel boundary components to an audience explicitly assumed to know them.

## 3\. Use one canonical name and statement for each object

Once an object is defined, subsequent sections should refer to its definition rather than redefining or paraphrasing it.

For example:

$$\mathfrak M_{\mathrm{Co},2}^{\mathrm{cov}},\qquad \mathfrak M_{\mathrm{Co},2}^{\mathrm{sing}},\qquad \mathcal F_{\mathrm{Co},2}^{\mathrm{Hdg}}$$

should each have one canonical definition. An undecorated $\mathcal F_{\mathrm{Co},2}$ should be introduced only after the equality of the relevant incarnations is proved, or under an explicit standing conjecture.

This is the practical value of Stacks-style stable references: one result is authoritative, and every later use points back to it. [![](https://www.google.com/s2/favicons?domain=https://stacks.math.columbia.edu&sz=128)The Stacks Project+1](https://stacks.math.columbia.edu/tags)

## 4\. Separate mathematical status from project-management status

Use ordinary mathematical environments consistently:

*   A proved assertion is a theorem, proposition, lemma, or corollary.

*   An expected exact assertion is a conjecture.

*   A genuinely exploratory request is a question.

*   A result conditional on unproved statements is a conditional theorem.

*   A finite calculation is a computational proposition.

*   A proposed but incompletely defined object is not a definition.

Do not write “Required theorem: prove $X=Y$.” Write:

> **Conjecture.** $X=Y$.

Then explain:

> Strategy A proves this conjecture by establishing Lemmas 4.1–4.3.

This preserves the mathematical information even if the research program changes.

## 5\. Introduce notation when it compresses structure

The correct standard is not “use as many symbols as possible.” It is:

> Give names to objects and maps that recur or participate in a diagram.

Good uses include:

*   parameter spaces $U_p$, $U$;

*   acting groups $G$, $G_p$;

*   period maps $\mathcal P_{\mathrm{dir}}$, $\mathcal P_{\mathrm{Hdg}}$;

*   normalization maps $\nu$;

*   reduction maps $r_k$;

*   cusp stabilizer homomorphisms;

*   classifying maps to KSBA moduli.

Avoid both extremes:

*   repeated prose such as “the image of the stabilizer of the root inside the degree-$2$ Enriques arithmetic group”;

*   dense decorations such as five nearly indistinguishable versions of $\Gamma$ used before their relations are established.

A notation table should record only globally recurring symbols, not every local variable.

## 6\. Begin each section with its role in the proof

A section opening should answer:

1.  What object is being studied?

2.  Why is it needed?

3.  What result will the section establish?

4.  Where will that result be used?

For example:

> This section computes the anti-invariant lattice of the Coble involution. The calculation determines the period domain in Definition 4.5 and supplies the embedded root complement used in the Heegner comparison of §8. The main result is Proposition 4.2.

This is substantially more useful than a generic section title followed immediately by formulas.

## 7\. State first, prove second, interpret third

For each mathematical unit:

1.  state the result;

2.  prove it;

3.  explain its role or consequence.

Do not mix the proof, caveats, anticipated applications, and historical mistakes into one paragraph.

A good local pattern is the one repeatedly used by Stacks: a brief transition, a precise result, and a proof whose sentences cite the exact prior definitions or lemmas used. [![](https://www.google.com/s2/favicons?domain=https://stacks.math.columbia.edu&sz=128)The Stacks Project](https://stacks.math.columbia.edu/tag/0DMV)

## 8\. Make proof sketches exact about what they omit

A proof sketch is acceptable when it gives a verifiable chain of reductions. It should say, for example:

> By Nikulin’s primitive embedding theorem, it suffices to identify the discriminant form and verify the stated parity invariant. Lemma 3.4 computes the form; the parity follows from the characteristic element $\eta/2$. Uniqueness then follows from \[exact theorem\].

It should not say:

*   “the result follows by separating”;

*   “the same argument works”;

*   “one checks”;

*   “standard transitivity gives”;

*   “the construction is natural”;

unless the relevant decomposition, check, theorem, or universal property is immediately specified.

Words such as “natural,” “canonical,” “standard,” and “compatible” are claims. Each requires a named category, action, or diagram relative to which the property holds.

## 9\. Allocate proof detail according to novelty and danger

Elementary local calculations can be concise for this audience. More detail should be devoted to:

*   integral versus rational lattice decompositions;

*   primitive closures and discriminant gluing;

*   descent of group actions to arithmetic quotients;

*   family-level simultaneous resolution;

*   extension of affine symmetries to algebraic involutions;

*   normalization through arithmetic and semitoroidal quotients;

*   finiteness and no-further-coarsening arguments.

These are precisely the places where a plausible sentence can conceal a false implication.

## 10\. Replace repeated warnings by exact hypotheses or obstruction lemmas

Use a warning only for a local and genuinely surprising exception.

Instead of:

> Warning: a wall slice is not a Coxeter diagram.

write:

> Proposition 6.7 constructs the sliced arrangement. To identify it with a  
> Coxeter chamber, it remains to prove:
> 
> 

$$\begin{aligned} &\text{(i) every displayed normal is a reflective root;}\\ &\text{(ii) the displayed roots are simple;}\\ &\text{(iii) their chamber is fundamental;}\\ &\text{(iv) every reflective wall meeting the chamber is displayed.} \end{aligned}$$

Instead of:

> Warning: the finite shadow is not the integral cusp.

define the reduction map and state the missing injectivity or lifting proposition.

Warnings should not be collected into a separate thirty-item section. The relevant hypothesis belongs next to the argument it controls.

## 11\. Remove internal history from the main line

The main text should contain the corrected theory. Preserve failed attempts only when they provide one of:

*   a counterexample;

*   a structural obstruction;

*   evidence discriminating between conjectures;

*   a reusable partial lemma;

*   a warning against a genuinely tempting false theorem.

Otherwise place them in a separate “Research history and discarded approaches” appendix. Each entry there should have the form:

> **Discarded claim.** …
> 
> **Failure.** The argument assumes …
> 
> **Correction.** Proposition $X$ replaces it.

Avoid descriptions such as “an earlier agent assumed” or general warnings against mistakes that no reader would otherwise make.

## 12\. Tie every computation to a mathematical proposition

Before a computation, state what it is intended to establish. Afterward, state exactly what follows.

A computational appendix should provide:

$$\text{input} \longrightarrow \text{algorithm} \longrightarrow \text{certificate} \longrightarrow \text{mathematical consequence}.$$

For orbit calculations, include group generators, representatives, invariants, orbit and stabilizer sizes, and the relation to the integral problem. For Gröbner calculations, include the ideals, charts, coefficient field, software, and script. For wall calculations, include the complete Gram matrix and the map from graph labels to lattice vectors.

Do not use anonymous labels such as $A,B,C,D$ in the main text unless they have intrinsic definitions.

## 13\. Formulate conjectures at their strongest justified level

When the discussions support an expected answer, state it. Do not replace

$$\Gamma_{\mathrm{Co},2}^{\mathrm{geom}} = \Gamma_{\mathrm{Co},2}^{\mathrm{dir}} = \Gamma_{\mathrm{Co},2}^{\mathrm{Hdg}} = \Gamma_{\mathrm{Co},2}^{\mathrm{cong}}$$

by “determine the relation between these groups.”

The equality is more informative and does not make the proof easier. It tells the reader what the theory predicts and permits conditional deductions. The proof program can then separate it into exact intermediate equalities or inclusions.

Use a question only when the expected answer is genuinely unknown:

> **Question.** Is the induced semifan rational polyhedral at cusp $c$, or does it have infinitely many rational extremal rays?

## 14\. Turn a strategy into a conditional implication

Each strategy section should contain:

*   a conditional theorem;

*   a proof of the conditional implication;

*   the exact unproved inputs;

*   the principal bottleneck;

*   comparison with other strategies.

It should not duplicate the entire document as a numbered list of things still to do.

For example:

> Strategy A reduces the comparison theorem to arithmetic normalization at the cusps and finiteness of the KSBA classifying map. Strategy B replaces arithmetic normalization by a direct recognizable-divisor theorem. The two strategies share the open-period theorem and integral cusp classification.

That comparison lets a researcher decide where effort is reusable.

## 15\. Use citations at the level of mathematical dependence

A citation should identify what is being imported.

Weak:

> “Standard transitivity results imply the following \[Wall; DK\].”

Better:

> “By Wall \[Theorem $X$\], $O(I_{2,9})$ acts transitively on primitive  
> isotropic vectors. Applying \[DK, Lemma $Y$\] to isotropic planes gives the  
> second assertion.”

For an adaptation:

> “The proof follows AEGS, Proposition $X$, except that the fixed exceptional  
> root changes the gluing condition in Step 3; this modification is proved in  
> Lemma $Y$ below.”

A broad source map may remain as an annotated bibliography, but it cannot replace local citations.