Generic Torelli for weighted hypersurfaces
We prove variational Torelli, hence generic Torelli, for a fairly wide class of hypersurfaces in weighted projective spaces. Namely, for hypersurfaces of degree d in \({\mathbb{P}}(1,1,q_ 1,...,q_ n)\) where the least common multiple m of the \(q_ i\) divides d and \(s=\sum q_ i+2,\quad and\) \(d\geq \max (3s,s+m(n+1)).\) The weighted case exhibits several complicating features which are completely absent in the projective case. These account for the incompleteness of our result and presumably also for the actual failure of Torelli for certain surfaces with \(p_ q=1\), some of which can be described as complete intersections of two weighted hypersurfaces. These new features include: (i) The general hypersurface of given degree and weights may be singular (quasi-smooth). - (ii) The two basic algebraic tools, Macaulay's theorem and the symmetrizer lemma, are false. This is ultimately traced back to the fact that the multiplication map between graded pieces in a weighted polynomial ring may not be surjective. - (iii) The existence of a coarse moduli space for weighted hypersurfaces is not clear, since the automorphism group of weighted projective space is not reductive.
hypersurfaces in weighted projective spaces
scientific article; zbMATH DE number 3944004
scientific article