Birational Mori fiber structures of ℚ-Fano 3-fold weighted complete intersections, II

In [T. Okada, Birational Mori fiber structures of \mathbb{Q} -Fano 3-fold weighted complete intersection, Proc. Lond. Math. Soc. (3) 109 2014, 6, 1549–1600], we proved that, among 85 families of \mathbb{Q} -Fano threefold weighted complete intersections of codimension two, 19 families consist of birationally rigid varieties and the remaining families consists of birationally non-rigid varieties. The aim of this paper is to study systematically the remaining families and prove that every quasismooth member of 14 families is birational to another \mathbb{Q} -Fano threefold but not birational to any other Mori fiber space.

Vector bundles on Fano threefolds of genus 7 and Brill–Noether loci

Let X be a smooth prime Fano threefold of genus 7 and let Γ be its homologically projectively dual curve. We prove that, for d ≥ 6, an irreducible component of the moduli scheme M X(2, 1, d) of rank-2 stable sheaves on X with c1 = 1, c2 = d is birational to a generically smooth (2d - 9)-dimensional component of the Brill–Noether variety [Formula: see text] of stable vector bundles on Γ of rank d - 5 and degree 5d-24 with at least 2d - 10 independent global sections. The space M X(2, 1, 6) is proved to be isomorphic to [Formula: see text], and to be a smooth irreducible threefold if X is general enough.

Simple Helices on Fano Threefolds

Building on the work of Nogin, we prove that the braid groupB4acts transitively on full exceptional collections of vector bundles on Fano threefolds withb2= 1 andb3= 0. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds withb2= 1 and very ample anticanonical class, every exceptional coherent sheaf is locally free.

Polarized endomorphisms of uniruled varieties. With an appendix by Y. Fujimoto and N. Nakayama

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on ℚ-Fano threefolds, Gorenstein log del Pezzo surfaces and ℙ1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree greater than one is rational.

Halphen pencils on weighted Fano threefold hypersurfaces

On a general quasismooth well-formed weighted hypersurface of degree Σi=14 a i in ℙ(1, a 1, a 2, a 3, a 4), we classify all pencils whose general members are surfaces of Kodaira dimension zero.