Singular del Pezzo fibrations and birational rigidity

A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of mobile divisors. The conjecture is proved to be true for smooth models (with a generality assumption for degree 3). It is speculated that the conjecture holds for, at least, Gorenstein models in degree 1 and 2. In this article, I present a (Gorenstein) counterexample in degree 2 to this conjecture.Comment: This is essentially a more detailed version of the second section of arXiv:1310.5548. To appear in the proceedings of the conference 'Groups of Automorphisms in Birational and Affine Geometry', held in Trento, Italy, 201

Non-rigid quartic 3-folds

Let X C P4 be a terminal factorial quartic 3-fold. If X is non-singular, X is birationally rigid, i.e. the classical minimal model program on any terminal Q-factorial projective variety Z birational to X always terminates with X. This no longer holds when X is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface X c P4. A singular point on such a hypersurface is either of type cAn (n > or equal 1), or of type cDm (m> or equal 4), or of type cE6, cE7 or cE8. We first show that if (P e X) is of type cAn, n is at most 7, and if (P \in X) is of type cDm, m is at most 8. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type cAn for 2\leq n\leq 7 (b) of a single point of type cDm for m= 4 or 5 and (c) of a single point of type cEk for k=6,7 or 8

Fano 3-folds in P2xP2 format, Tom and Jerry

We study Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of P^2xP^2. These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classification in three different ways. Some families arise as unprojections of degenerations of complete intersections, where the generic unprojection is a known prime Fano 3-fold in codimension 3; these are new, and an analysis of their Gorenstein projections reveals yet other new families. Others represent the "second Tom" unprojection families already known in codimension 4, and we show that every such family contains one of our models. Yet others have no easy Gorenstein projection analysis at all, so prove the existence of Fano components on their Hilbert scheme