Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to $\mathbb P^3$

Abstract

We study well-formed weighted projective threefolds that admit $\mathbb Q$-Gorenstein smoothings to $\mathbb P^3$. Two families are known: the $\mathbb P^2$-type and the $Q$-type, and it is conjectured that these are the only possibilities. We derive numerical and local necessary conditions for such a smoothing. In addition to the anticanonical volume equation, constancy of the anticanonical Hilbert polynomial yields a further identity when all codimension two singularities are of $A$-type. We also obtain a semigroup condition governing the existence of global smoothing directions along codimension two curves with transverse $A$-type singularities. We apply these conditions to prove the expected classification in several cases. In particular, for every fixed square-free integer $d$, there are only finitely many $\mathbb Q$-Gorenstein smoothable spaces $\mathbb P(1,a,b,c)$ such that $\gcd(a,b)=d$. Our method reduces the possible weights to a finite exact computation; for every prime $p\le100$, the computation produces only members of the two expected families. Finally, we prove the classification when $\gcd(a,b)=d$ and $a=d^2$.