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.
For an arbitrary ample divisor A in smooth del Pezzo surface S of degree 1, we verify the condition of the polarization (S,A) to be K‐stable and it is a simple numerical condition.
We prove the factoriality of every nodal quartic threefold with 13 singular points that contains neither planes nor quadric surfaces. As a corollary, any nodal quartic threefold with 13 singular points that contains neither planes nor quadric surfaces is nonrational.
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