Fano 3-folds in format, Tom and Jerry

We study $${\mathbb {Q}}$$Q-factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of . 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.

[1]  J. Chen,et al.  On Quasismooth Weighted Complete Intersections , 2009, 0908.1439.

[2]  Enrico Fatighenti,et al.  Hodge Numbers and Deformations of Fano 3-Folds , 2017, Documenta Mathematica.

[3]  B. Szendrői,et al.  Constructing projective varieties in weighted flag varieties , 2010, 1008.1947.

[4]  M. Reid,et al.  Weighted Grassmannians , 2002, math/0206011.

[5]  M. Reid.  Gorenstein in codimension 4 - the general structure theory , 2013, 1304.5248.

[6]  M. I. Qureshi,et al.  Calabi-Yau Threefolds in Weighted Flag Varieties , 2011, 1105.4282.

[7]  Gavin Brown,et al.  Graded rings and special K3 surfaces , 2006 .

[8]  Carmelo Di Natale,et al.  Hodge theory and deformations of affine cones of subcanonical projective varieties , 2015, J. Lond. Math. Soc..

[9]  Miles Reid,et al.  Explicit Birational Geometry of 3-Folds: Fano 3-fold hypersurfaces , 2000 .

[10]  N. Ilten.  Versal deformations and local Hilbert schemes , 2011, 1107.2416.

[11]  Hiromichi Takagi.  On classification of ℚ-fano 3-folds of Gorenstein index 2. II , 2002, Nagoya Mathematical Journal.

[12]  Kei-ichi Watanabe,et al.  On graded rings, I , 1978 .

[13]  A. Kustin,et al.  Constructing big Gorenstein ideals from small ones , 1983 .

[14]  M. Reid,et al.  Fano 3-folds in codimension 4, Tom and Jerry. Part I , 2010, Compositio Mathematica.

[15]  M. Reid,et al.  Diptych varieties, I , 2012, 1208.2446.

[16]  A. Kasprzyk,et al.  Gorenstein Formats, Canonical and Calabi–Yau Threefolds , 2014, Exp. Math..

[17]  Miles Reid,et al.  Kustin–Miller unprojection without complexes , 2004 .

[18]  M. I. Qureshi.  Constructing projective varieties in weighted flag varieties II , 2014, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Gavin Brown,et al.  A Database of Polarized K3 Surfaces , 2007, Exp. Math..

[20]  Hamid Abban.  On pliability of del Pezzo fibrations and Cox rings , 2013, 1304.4357.

[21]  A. R. Iano-Fletcher,et al.  Explicit Birational Geometry of 3-Folds: Working with weighted complete intersections , 2000 .

[22]  Muhammad Imran Qureshi,et al.  Computing isolated orbifolds in weighted flag varieties , 2015, J. Symb. Comput..

[23]  M. Reid,et al.  Ice cream and orbifold Riemann–Roch , 2012, 1208.0457.