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Abstract:
It is believed that smooth, non complete intersection, codimension 2 subvarieties
of Pn become more rare when n increases. In a joint work with Ellia and
Franco, we prove the
following result in this direction: Let X be a smooth codimension 2 subvariety of P5,
of degree d, lying on a
hypersurface of degree s.
Let g be the genus of the intersection curve of X with a general P3
in P5. One has the
inequality
d(s^2 -4s+d) - s(2g-2) ≤ s(s-1)^3
which implies (unfortunately in a non-explicit way) that for fixed s ,
the set of "liaison"
classes of those X is limited (that is, can be parametrized by some
algebraic variety). For n>5
the set of those X which are not complete intersections is itself
limited, in fact we give an
explicit upper bound on d.
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