On factorial double solids with simple double points

Abstract

We study when double covers of P 3 ramified along nodal surfaces are not Q-factorial. In particular, we describe all the Qfactorial double covers of P 3 ramified along quartic surfaces with at most seven simple double points and sextic surfaces with at most 16 simple double points.

Points in projective spaces and applications

“…• the threefold X is not factorial, which implies that there is a point P ∈ Σ such that every surface in P 3 of degree 3r − 4 containing Σ \ P contains the point P . We assume that r 4, because the case r = 3 is done in [11]. Lemma 3.5.…”

2 n2-inequality for cA1 points and applications to birational rigidity

“…The Q-factoriality of nodal double solids has been studied extensively in [CP10,Che09,HP07,Klo22]. Their results are mainly based on the result of [Cle83].…”

“…As mentioned previously, in this section, we use the result of [Ram08] that is applicable to the cases with wider range of singularities. Before we proceed, let us mention that some results in [CP10,Che09,HP07,Klo22] remain true if we allow only singularities slightly worse than nodes.…”

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity

“…Q-factoriality of nodal double solids has been extensively studied in [CP10], [Che09], [HP07] and [Klo22]. Their results are mainly based on the result of [Cle83].…”

“…As aforementioned, in this section, we use the result of [Ram08] that is applicable to the cases with wider range of singularities. Before we proceed, let us mention that some results in [CP10], [Che09], [HP07] and [Klo22] remain true if we allow only singularities slightly worse than nodes.…”