We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial Ga or Gm-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.

Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a notion originally related to the action of differential operators on the polynomial ring. In this work, we make explicit how one can see apolarity in terms of the Cox ring of a variety. In this way, powersum varieties are a special case of varieties of apolar schemes; we explicitly describe examples of such varieties in the case of two toric surfaces, when the Cox ring is particularly well-behaved.

In this note we show that if a projective manifold admits a Khler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.