Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

Un classico teorema di G. Gherardelli afferma che una curvaC P 3 è intersezionè completa se e soltanto se è proiettivamente normale e sottocanonica. Qui si prova che, seC e a- sottocanonica ed inoltre le superficie di grado 1 + (a/2) (a pari) ovvero (a + l)/2 o (a + 3)/2 o (a + 5)/2 (a dispari) tagliano suC serie complete, alloraC è intersezione completa. Si determina inoltre un bound d funzione di a tale che, seC è a-sottocanonica e di grado dles d, alloraC è intersezione completa se e soltanto se le superficie di grado a tagliano suC una serie completa. Si discutono poi numerosi esempi di curve sottocanoniche non intersezioni complete

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Let E be an indecomposable rank two vector bundle on the projective space P n , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Qn ⊂ P n+1 , n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q3, k ≥ 2, we prove two boundedness results.Keywords: arithmetically Buchsbaum rank two vector bundles, smooth quadric hypersurfaces. MSC 2010: 14F05.Theorem. Let E be an arithmetically Buchsbaum, normalized, rank 2 vector bundle on the projective space P n , n ≥ 3. Then E is one of the following: 1. n ≥ 3 : E is a split bundle; 2. n = 3 : E is stable with c 1 = 0, c 2 = 1, i.e. E is a null-correlation bundle. * The paper was written while all authors were members of INdAM-GNSAGA. Lavoro eseguito con il supporto del progetto PRIN "Geometria delle varietà algebriche e dei loro spazi di moduli", cofinanziato dal MIUR (cofin 2008).

In [2], chap. IV, 2me partie, (7.4.8), Grothendieck considered the following problem: is any m-adic completion of an excellent ring A also excellent?In [8] I proved that, if A is an algebra of finite type over an arbitrary field k, the answer is positive.