Divisorial Mori Contractions of Submaximal Length

A result due to Cho, Miyaoka, and Shepherd-Barron [in: Higher dimensional birational geometry (2002), 1–88] and Kebekus [in: Complex geometry (2002), 147–155] provides a numerical characterization of projective spaces. More recently, Dedieu and Höring [Algebr. Geom. 4 (2017), 120–135] gave a characterization of smooth quadrics based on similar arguments. As a relative version of [in: Higher dimensional birational geometry (2002), 1–88] and Kebekus [in: Complex geometry (2002), 147–155], Höring and Novelli [Publ. Res. Inst. Math. Sci. 49 (2013), 215–228] proved that the locus covered by positive-dimensional fibers in a Mori contraction of maximal length is a projective bundle up to birational modification. We change the length hypothesis and we prove that the exceptional locus of a divisorial Mori contraction of submaximal length is birational either to a projective bundle, or to a quadric bundle.