Sharp factoriality threshold and defect-one equality cases

Definitions and mathematical statement

Let $X_n\subset{\mathbb{P}}^4$ be a degree-$n$ hypersurface with a finite set $\Sigma=\operatorname{Sing}X_n$ of ordinary double points and no other singularities. In the attached aid the field is ${\mathbb{C}}$; the extracted original does not state it. An ordinary double point is a hypersurface singularity with nondegenerate quadratic part in the intended characteristic-zero setting. Write $\operatorname{Cl}(X_n)$ for Weil divisors modulo linear equivalence. The source uses "factorial" to mean $\operatorname{Cl}(X_n)={\mathbb{Z}}$, with the hyperplane class understood as generator.

Original questions (Cheltsov). Determine when $X_n$ is factorial. In particular, does \[|\Sigma|<(n-1)^2\quad\Longrightarrow\quad\operatorname{Cl}(X_n)={\mathbb{Z}}?\] Also, for a generic linear projection $\phi:{\mathbb{P}}^4\dashrightarrow{\mathbb{P}}^2$, is it true that, for every integer $k\ge1$ and every plane curve $B$ of degree $k$, \[|\phi(\Sigma)\cap B|\le k(n-1)?\] The projection should be defined and injective on the finite set $\Sigma$, as occurs for a sufficiently general center. The extracted projection question states no cardinality hypothesis on $\Sigma$; this must not be silently replaced by the more restricted version discussed by the aid.

The original attributes to Clemens a criterion using independent conditions on degree-$n$ forms. The aid instead gives degree $2n-5$: \[X_n\text{ factorial}\quad\Longleftrightarrow\quad H^0({\mathbb{P}}^4,\mathcal O(2n-5))\longrightarrow H^0(\Sigma,\mathcal O_\Sigma(2n-5))\text{ is surjective}.\] This correction is explicitly attributed to the aid; the inconsistent extracted degree is retained as a source issue. One may measure the indicated defect by $\delta(X_n)=h^1({\mathbb{P}}^4,\mathcal I_\Sigma(2n-5))$.

The displayed source example, in coordinates $x,y,z,w,v$, is \[X_n=\{xf_{n-1}+yg_{n-1}=0\},\] where $f_{n-1},g_{n-1}$ are generic homogeneous forms. It reports $\operatorname{Cl}(X_n)={\mathbb{Z}}\oplus{\mathbb{Z}}$. In the intended $n\ge2$ nodal example the hypersurface contains the plane $x=y=0$; its node set in the equality case is of complete-intersection type $(1,1,n-1,n-1)$.

Two extracted "known facts" require caution. The first has a damaged coefficient printed as $2$, an intervening comparison sign, and $3$ before $(n-1)^2$, so its numerical hypothesis cannot be certified from this record. The second says that every smooth codimension-one subvariety $S\subset X_n$ is Cartier. Its direction or omitted qualification needs checking: taken literally it conflicts with the role of the smooth contained plane in the displayed nonfactorial example. Neither damaged claim is accepted here as a verified theorem.

Unreviewed research-aid claims. The aid attributes to Cheltsov the sharp affirmative threshold over ${\mathbb{C}}$: every nodal degree-$n$ hypersurface with at most $(n-1)^2-1$ nodes is factorial. It says that under the exact Cheltsov--Park cardinality hypothesis $|\Sigma|\le(n-1)^2-1$, the full generic-projection incidence assertion was not found resolved, while the required bound was proved for $k\le\sqrt{n-1}$. This is a report about that restricted formulation and literature search, not an assertion that the unrestricted extracted question has the same status.

Finally, the aid claims that every nonfactorial complex nodal degree-$n$ hypersurface with exactly $(n-1)^2$ nodes has reduced node scheme a complete intersection of type $(1,1,n-1,n-1)$, defect $\delta(X_n)=1$, and $\operatorname{Cl}(X_n)\cong{\mathbb{Z}}^2$, not just generic hypersurfaces of the displayed equation. Its necessary degree range, reference and full proof are not supplied. These assertions require independent mathematical and source review.

Short English statement

Cheltsov asks for factoriality and a generic-projection incidence bound for nodal threefold hypersurfaces. The aid reports the sharp node-count threshold and a defect-one equality claim. The draft flags the conflicting criterion degrees, damaged historical bound, questionable smooth-divisor statement, and the cardinality hypothesis present only in the aid's projection discussion.

Sources

• Local UnsolvedMath registry, permanent ID 20000318, source code catalogue entry; aim-algebraic-geometry-notes.json, notes[317], tag problem.
• AIM workshop Numerical invariants of singularities and higher-dimensional algebraic varieties, Problem 8 (Cheltsov): https://aimath.org/WWN/singularvariety/problems_final.pdf. The original names Clemens. The supplied status is reconstructed_unverified; its verification field refers to original TeX but supplies no complete repaired transcription. No PDF or TeX source was consulted here.
• Attached research attempt 1 names Cheltsov and Cheltsov--Park without bibliographic locators; original literature field empty. Result partial_result, status partially_solved, rigor label proved, novelty confidence low. It is a machine-generated, non-peer-reviewed aid requiring independent verification before citation.