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Sur les surfaces lisses de $${\mathbb{P}}_ 4$$. (On the smooth surfaces of $${\mathbb{P}}_ 4)$$. (French) Zbl 0676.14009
Let $${\mathbb{P}}^ 4$$ be the four-dimensional projective space over an algebraically closed field of characteristic zero. The authors prove that the smooth algebraic surfaces S in $${\mathbb{P}}^ 4$$ satisfying the inequality $$K^ 2_ S\geq a\chi ({\mathcal O}_ S)$$ for $$a\in {\mathbb{R}}$$ and $$a<6$$, are distributed in finitely many components of the Hilbert scheme of the smooth algebraic surfaces of $${\mathbb{P}}^ 4.$$
It results as a corollary that the smooth rational surfaces of $${\mathbb{P}}^ 4$$ describe finitely many components of the Hilbert scheme, as conjectured by Hartshorne and Lichtenbaum.
Reviewer: E.Casas-Alvero

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14C05 Parametrization (Chow and Hilbert schemes) 14N05 Projective techniques in algebraic geometry
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##### References:
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