Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to Fano threefolds with Picard number one

Abstract

We study well-formed weighted projective threefolds $X=\mathbb P(w_0,w_1,w_2,w_3)$ that admit a $\mathbb Q$-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds $V_d$ and prime Fano threefolds $Y_g$, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal $A$-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for $V_5$, with $w_0+w_1+w_2+w_3\leq 5000$. We construct $\mathbb Q$-Gorenstein smoothings to $V_1, V_2, Y_6, Y_{10},$ and $Y_{12}$ from numerical candidates of weighted projective threefolds, and prove that $\mathbb P(2,5,8,25)$, although a numerical candidate for $V_4$, is not $\mathbb Q$-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.