Generic Torelli for weighted hypersurfaces
type: Item
instance of: scholarly article
title: Generic Torelli for weighted hypersurfaces
zbMATH Open document ID: 0588.14003
DOI: 10.1007/BF01450837
author: Loring W. Tu
author: Ron Donagi
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164207
review text: 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.
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 3944004
zbMATH Keywords: variational Torelli
zbMATH Keywords: generic Torelli
zbMATH Keywords: hypersurfaces in weighted projective spaces
zbMATH Keywords: failure of Torelli
MaRDI profile type: Publication
cites work: Q3739270
cites work: Variational Torelli implies generic Torelli
cites work: Espaces projectifs anisotropes
cites work: Q3731740
cites work: Q3674803
cites work: Q5735629
cites work: Weak global Torelli theorem for certain weighted projective hypersurfaces
cites work: Q3749165
OpenAlex ID: W1990705547
description: scientific article; zbMATH DE number 3944004