On Fano Schemes of Complete Intersections

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 319)


We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain projective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.


Hypersurfaces Complete intersections Fano schemes 

2010 Mathematics Subject Classification

Primary: 14J70 14M10 14N10 14N15 Secondary: 14C05 14C15 



The first author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. He also thanks the GNSAGA of INdAM. This research was partially done during a visit of the second author at the Department of Mathematics, University of Rome Tor Vergata (supported by the project“Families of curves: their moduli and their related varieties”, CUP E8118000100005, in the framework of Mission Sustainability). The work of the second author was also partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund (code: BCSim-2018-s09). The authors thank all these Institutions and programs for the support and excellent working conditions.

Our special thanks are due to Laurent Manivel who kindly suggested to complete some references omitted in the first draft of this paper. We are grateful to the peer reviewer whose suggestions allowed to improve the style and to avoid some typos and arithmetical mistakes.


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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di Roma “Tor Vergata”, Via della Ricerca ScientificaRomeItaly
  2. 2.CNRS, Institut Fourier, Univ. Grenoble AlpesGrenobleFrance

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