Global (log) canonical threshold of d-sheeted covers of M-dimensional projective space of index 1 is 1 for almost all families (d≥4) under quadratic singularity and regularity assumptions, implying birational rigidity of Fano-Mori fibre spaces with base dimension bounded quadratically by fibre dim.
Birationally rigid finite covers of the projective space
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abstract
In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$ and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant 7$, satisfying certain regularity conditions. Up to now, only cyclic covers were studied in this respect. The set of varieties with worse singularities or not satisfying the regularity conditions is of codimension $\geqslant\frac12(M-4)(M-5)+1$ in the natural parameter space of the family.
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Canonical and log canonical thresholds of multiple projective spaces
Global (log) canonical threshold of d-sheeted covers of M-dimensional projective space of index 1 is 1 for almost all families (d≥4) under quadratic singularity and regularity assumptions, implying birational rigidity of Fano-Mori fibre spaces with base dimension bounded quadratically by fibre dim.