2022/07/14 by Stephen Coughlan, Coughlan, Stephen, Roberto Pignatelli +1 · 1 citation
Mathematics · #14E30 #14J15 #14J29 #14J30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Canonical bundle #Commutative Algebra and Its Applications #FOS: Mathematics #Fibration #Geometry and complex manifolds #Gravitational singularity #Hypersurface #K3 surface #Line bundle #Mathematical analysis #Mathematics #Moduli space #Noether's theorem #Projective line #Projective space #Projective test #Pure mathematics #Simple (philosophy)
paper · pdf · doi:10.48550/arxiv.2207.06845
openalex publication_date 2022/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce the notion of a simple fibration in (1,2)-surfaces. That is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with pg=2 and K2=1. We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds "on the Noether line" with K3=\frac43 pg-\frac103, and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over ℙ1 are Cartier divisors in a toric 4-fold. This allows to us to show among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space, and that there is a second component when the geometric genus is congruent to 6 modulo 8; the threefolds in this component are new.