Classification conjecture for weighted projective spaces admitting Q-Gorenstein smoothings to P3

The conjecture asks whether all weighted projective spaces admitting a $\mathbb{Q}$-Gorenstein smoothing to $\mathbb{P}^{3}$ belong to two explicit families. The expected classification is not known in general.

Known results

• Hacking and Prokhorov classified the analogous surface smoothings, giving the Markov-type equation $3abc=a^{2}+b^{2}+c^{2}$.
• Chen and Lee construct two infinite smoothable families in higher dimensions.
• A bounded computation lists $23$ candidate four-dimensional weight tuples with weights at most $800$, but does not prove completeness.
• Some smoothable examples lie outside the two principal families.

September 2026 special-case result

Chen and Lee report a proof when $\gcd(a,b)=d$ and $a=d^{2}$, together with finite exact computations for fixed square-free $d$ and primes through $100$. This narrows the classification but does not settle the remaining cases; the reported advance is not independently verified here.

Current status (as of September 2026): The two-family classification remains open in general; the case $\gcd(a,b)=d$ with $a=d^{2}$ is claimed to be proved, but that claim is unverified.