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SUMMARY:Fabrizio Catanese (University of Bayreuth)
DTSTART:20210128T140000Z
DTEND:20210128T150000Z
DTSTAMP:20260423T052838Z
UID:ZAG/90
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ZAG/90/">Var
 ieties of nodal surfaces\, Coding theory\, and cubic discriminants</a>\nby
  Fabrizio Catanese (University of Bayreuth) as part of ZAG (Zoom Algebraic
  Geometry) seminar\n\n\nAbstract\nNodal Hypersurfaces Y  in projective spa
 ce are those  whose singularities have nondegenerate Hessian. \nBasic nume
 rical invariants are the dimension n and  the degree d of the  hypersurfac
 e Y\, and  the number \\nu of singular points. \nIf you fix those integers
  (n\, d\,\\nu) these hypersurfaces  are parametrized by the so-called Noda
 l Severi varieties F(n\, d\, \\nu).  \nThe first basic questions concernin
 g them are: \n1) for which triples  is F(n\, d\, \\nu) nonempty ? \n2) Whe
 n is it irreducible ? \n\nAlready intriguing is the situation for surfaces
 \, indeed for n=2 the answer to 1) is known for d <= 6\, also the maximal 
 number of nodes \\mu (d)  that a nodal surface in 3-space of degree d can 
 have is known only  for d <= 6.\n\nThe known maximizing nodal surfaces (th
 ose with \\mu(d) nodes) are: the Cayley cubic\, the Kummer quartic surface
 s\, the Togliatti quintics\, the Barth sextic.\n\nAn important chapter in 
 Coding theory is the theory of binary linear codes\, vector subspaces of a
  vector space  (Z/2)^n.\n\nI will recall basic notions and methods of codi
 ng theory (e.g. the McWilliams identities) and describe some  codes relate
 d to quadratic forms. \n\nNodal surfaces are related to coding theory via 
 the first homology of their smooth part: it  is a binary code K\, which wa
 s used  by Beauville to show that\, for d=5 \, \\mu(d) = 31. Coding theory
  was also crucial in order to prove that \\mu(6) < = 65. \n\nOur main resu
 lts concern the cases d = 4\,5\,6 (d=2\,3 being elementary). \n\nTHM 1. Fo
 r d=4 the components of  F(4\, \\nu) and their incidence correspondence ar
 e determined by their extended codes K’\, which  are all the shortenings
  of the first Reed Muller code.\n\nWe extend this result to nodal K3 surfa
 ces of all degrees\, this sheds light on the  case  d=5.\n\nTHM 2. For d=5
    the codes K occurring are classified\, up to a possible exception. F(5\
 , \\nu) is irreducible for \\nu = 31\, and for \\nu = 29\,30\,31 these sur
 faces are discriminants   of the projection of a cubic hypersurface in 5-s
 pace. \n\nTHM 3. For d=6 and \\nu = 65  the codes K\, K’ are  uniquely d
 etermined\, and can be described explicitly via  the Doro-Hall graph\, att
 ached to the group \\SigmaL(2\, 25)\, and the geometry of the Barth sextic
 . Every 65 nodal sextic occurs as  discriminant of the projection of a cub
 ic hypersurface in  6-space  with < = 33  nodes.\n\n\nIrreducibility for d
 =6\, and 65 nodes\, is related to  the geometry of nodal cubic hypersurfac
 es in n-space\, and of the linear subspaces contained in them.\n\nWe pose 
 the question  whether\, in the case of even dimension n\, the cubic hypers
 urface with maximal number of singularities is\nprojectively equivalent to
  the Segre cubic s_1=s_3=0 (which is locally rigid).\n\nFor theorem 2  I b
 enefited of the  cooperation of Sandro Verra\, for theorem 3 of Yonghwa Ch
 o\, Michael Kiermaier\,  Sascha Kurz and the Linux Cluster of the Universi
 taet Bayreuth\,  while  Davide Frapporti and Stephen Coughlan cooperated f
 or the geometry of nodal cubic hypersurfaces.\n
LOCATION:https://researchseminars.org/talk/ZAG/90/
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