Known irreducibility regions and the explicit q=1 component decomposition

Definitions and mathematical statement

The report recovers $E_{p,q,r}$ as a based representation scheme for the polynomial ring in four variables. Over an algebraically closed field $k$, fix graded vector spaces $M_0=k^p$, $M_1=k^q$, $M_2=k^r$, with all other degrees zero. The four multiplication maps in each adjacent degree are \[ A_i:k^p\to k^q,\qquad B_i:k^q\to k^r\quad(1\leq i\leq4). \] Then \[ E_{p,q,r}=\{(A_i,B_i)_{i=1}^4:B_iA_j=B_jA_i\text{ for }1\leq i<j\leq4\} \] is an affine scheme defined by six matrix commutativity relations. This is the scheme with bases fixed, not an unspecified quotient by changes of basis.

Problem 10. For which Hilbert functions $\rho=(p,q,r,0,\ldots)$ is $E_{p,q,r}=E_\rho$ irreducible? The original extract calls this a moduli space without supplying the four-variable or based-scheme convention; the report's recovery makes those choices explicit.

Original remark and reported correction. The source credits Martin--Deschamps--Perrin with reducibility when \[ 4q<\max\{6p+r,6r+p\}, \] and conjectures irreducibility when the reverse weak inequality holds. As written, the first statement misses an exception: the aid says $(p,q,r)=(1,1,1)$ is irreducible although $4<7$. The exceptional triple must therefore be restored before quoting that reducibility criterion for positive triples. The aid also says Ginouillac's later results refine the known/open region, but their inequalities and hypotheses are not included in the supplied summary, so the full modern classification cannot be reconstructed here.

Reported complete $q=1$ decomposition. Let $p,r>0$, $q=1$. Regard $A_i$ as row vectors of length $p$ and $B_i$ as column vectors of length $r$. Form the $(p+r)\times4$ matrix \[ M(A,B)=\begin{pmatrix}A_1^{\mathsf T}&A_2^{\mathsf T}&A_3^{\mathsf T}&A_4^{\mathsf T}\\ B_1&B_2&B_3&B_4\end{pmatrix}. \] The aid proves the reduced-scheme decomposition \[ (E_{p,1,r})_{\mathrm{red}} =\{A_1=\cdots=A_4=0\}\ \cup\ \{B_1=\cdots=B_4=0\} \ \cup\ \{\operatorname{rank}M(A,B)\leq1\}. \] The first two loci are affine linear spaces; the third is the irreducible rank-at-most-one determinantal variety. Removing those loci contained in others gives exactly \[ \#\{\text{irreducible components of }E_{p,1,r}\} =\begin{cases} 1,&p=r=1,\\ 2,&\text{exactly one of }p,r\text{ equals }1,\\ 3,&p>1\text{ and }r>1. \end{cases} \] This statement is over every algebraically closed field as reported. It determines the irreducible supports and does not assert that the original commutativity-equation scheme is reduced. It addresses the middle-degree-one slice, not all positive triples or unspecified zero-dimensional-degree edge cases.

Short English statement

Classify irreducibility of the based four-variable graded-module scheme with dimensions $(p,q,r)$. The aid corrects an omitted $(1,1,1)$ exception in the original remark and gives the full reduced decomposition when $q=1$: two zero-map loci together with one rank-one determinantal locus yield one, two or three components.

Sources

• Local UnsolvedMath registry, permanent ID $20000811$, catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[56].
• AIM workshop "Components of Hilbert schemes," PDF Problem 10 and its nested remark: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf.
• M. Martin--Deschamps, D. Perrin, "Courbes gauches et modules de Rao," J. Reine Angew. Math. 439 (1993), 103--145, p. 119, Theorem 2.1, as cited in the original remark. The report also names later work of Ginouillac without a complete citation or statement.
• Research attempt 1, partial_result, run status open, internal rigor proved, low novelty confidence; current registry partially solved. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.