We prove a structural result for geometrically non-reduced varieties and give applications to Fano varieties. For example, we show that if
X
X
is the generic fibre of a Mori fibre space of relative dimension
n
n
, and the characteristic is
p
>
2
n
+
1
p>2n+1
, then any geometric non-reducedness of
X
X
comes from the base of some fibration.
We prove a structural result for geometrically non-reduced varieties and give applications to Fano varieties. For example, we show that if X is the generic fiber of a Mori fibre space of relative dimension n, and the characteristic is p > 2n + 1, then any geometric non-reducedness of X comes from the base of some fibration.
Let T be a complete equicharacteristic local (Noetherian) UFD of dimension 3 or greater.Assuming that |T | = |T /m|, where m is the maximal ideal of T , we construct a local UFD A whose completion is T and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, T is regular and has characteristic zero, we can construct A to be excellent.
We study real double covers of $\mathbb P^{1}\times \mathbb P^{2}$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $\mathbb R$-rational. For five of the six isotopy classes, we construct $\mathbb C$-rational examples with obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors.
We study real double covers of P 1 ×P 2 branched over a (2, 2)-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct C-rational examples that have obstructions to rationality over R, and for the sixth class, we show that the models we consider are all rational.Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett-Tschinkel and Benoist-Wittenberg. The double cover models we consider were introduced and previously studied by S.
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