Sur les surfaces lisses de \({\mathbb{P}}_ 4\). (On the smooth surfaces of \({\mathbb{P}}_ 4)\) (Q1122632)

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.