Explicit diagonal 1-PS bounds and the current hypersurface GIT classification status

Definitions and mathematical statement

Determine the GIT stable and semistable degree-five surfaces in ${\mathbb{P}}^3$, degree-four threefolds in ${\mathbb{P}}^4$, and degree-five threefolds in ${\mathbb{P}}^4$. Also, for fixed hypersurface dimension and degree, bound the exponents of diagonal one-parameter subgroups needed for testing. The original says smaller cases had been checked by various authors, without references.

In the usual complex GIT interpretation, use the $\operatorname{SL}_{N+1}$ action and standard linearization on ${\mathbb{P}}(\operatorname{Sym}^d({\mathbb{C}}^{N+1})^*)$. A diagonal $1$-PS has integer weights $a=(a_0,\ldots,a_N)$, $\sum_i a_i=0$, and is primitive if their gcd is one. Under the chosen coordinate-weight convention, a monomial $x^\alpha$ of total degree $d$ has weight $a\cdot\alpha$. With $w_a(F)=\min_{c_\alpha\ne0}a\cdot\alpha$, semistability requires $w_a(gF)\le0$ and stability requires $w_a(gF)<0$ for every nontrivial $a$ and every required change of coordinates $g$. Thus weak witnesses $w_a\ge0$ detect failure of stability, while strict witnesses $w_a>0$ detect instability. Bounds on diagonal weights do not remove the coordinate-change quantifier.

The attached unreviewed aid proposes \[ D_{N,d}=\left\lceil\sqrt N\,d^{N-1}\right\rceil,\qquad B_{N,d}=N D_{N,d}. \] It claims every weak Hilbert--Mumford witness can be replaced, after the required coordinate change, by a primitive diagonal witness with $\max_i|a_i|\le D_{N,d}$. It further claims every diagonal $1$-PS has a primitive representative with identical signs on all degree-$d$ monomial weights and $\max_i|a_i|\le B_{N,d}$, giving the latter bound for strict witnesses. Its reported stability/semistability bounds are respectively $44/132$ for quintic surfaces, $128/512$ for quartic threefolds and $250/1000$ for quintic threefolds.

The aid reports algorithmic finite-state descriptions for all three cases but says their full geometric interpretation remains open or at announcement level in important parts. This is the supplied literature assessment, not a fresh status determination. The exact algorithms, classification strata and proofs of the proposed bounds are not supplied; numerical witness bounds alone do not classify the geometric stable and semistable loci.

Short English statement

Classify the three specified hypersurface GIT loci and bound the one-parameter subgroup weights needed for testing. The aid gives separate proposed bounds for weak and strict witnesses, while retaining all coordinate changes and the missing geometric classification.

Sources

• Problem 3.2, 2004 AIM Compact moduli spaces and birational geometry: https://aimath.org/WWN/birational/birational.pdf. Canonical location: aim-algebraic-geometry-notes.json, notes[332]. Supplied verification says the PDF and TeX agree with the record; no source visit or PDF compilation is asserted here.
• The original mentions unspecified authors for smaller cases. The attached aid reports algorithms and partial geometric interpretations without names or full bibliographic entries, and proposes the weight bounds with low novelty confidence. It is machine generated and not peer reviewed.
• Local registry: data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry.