Non-quasi-projective moduli spaces. (English) Zbl 1140.14011

A polarized variety is a pair \((X, H)\) consisting of a smooth projective variety and a linear equivalence class of ample divisors \(H\) on it. For simplicity, take \(X\) smooth, \(H\) very ample and \(H^i(X, \mathcal{O}_X(m H)) = 0\) for \(i, m > 0\). A well known way to construct moduli spaces of such pairs is to embed \(X\) into \(\mathbb{P}^N\) by \(| H| \). Then one can view the quotient \(U(X, H)/PGL(N + 1)\), where \(U(X, H)\) is an open set (of the deformations of \((X, H)\)) of the Hilbert scheme Hilb \((\mathbb{P}^N)\) with Hilbert polynomial \(\mathcal{X}(X, \mathcal{O}_X(mH))\), as the moduli space of the pairs \((X, H)\). Usually, one requires that the action be proper and this is equivalent to assuming that the quotient \(U(X, H)/PGL(N + 1)\) exists as a separated complex space or as a separated algebraic space.
In the paper under review, the author shows that every smooth toric variety and many other algebraic spaces can be realized as moduli spaces for smooth, projective, polarized varieties. Some of these are not quasi-projective: the examples (9) and (29) show that the quotients \(U(X, H)/ PGL(N + 1)\) can contain smooth, proper subschemes which are not projective. This contradicts the main result of a recent paper [G. Schumacher, H. Tsuji, Ann. Math. 159, No. 2, 597–639 (2004; Zbl 1068.32011)].


14D20 Algebraic moduli problems, moduli of vector bundles
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
32G05 Deformations of complex structures


Zbl 1068.32011
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