arXiv subjects
Publications and source records attributed to Paolo Valabrega.
The paper proves the intermediate value theorem for polynomials and power series over a valued field with divisible valuation group and infinite residue field. Some further results on the behaviour of the valuation are obtained using Hensel's Lemma.
In the present paper we investigate the convergence of a double series over a complete non-Archimedean field and prove that, while the proofs are somewhat different, the Archimedean results hold true.
Let $E$ be an indecomposable rank two vector bundle on the projective space $\PP^n, n \ge 3$, 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 $Q_n\subset\PP^{n+1}$, $n\ge 3$. 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 $Q_3$, $k\ge2$, we prove two boundedness results.
The paper investigates the non-vanishing of $H^1(E(n))$, where $E$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that $Pic(X) \cong \ZZ$. If $ε$ is the integer defined by the equality $ω_X = O_X(ε)$, and $α$ is the least integer $t$ such that $H^0(E(t)) \ne 0$, then, for a non-stable $E$ ($α\le 0$) the first cohomology module does not vanish at least between the endpoints $\frac{ε-c_1}{2}$ and $-α-c_1-1$. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on $α$ and also on the second Chern class $c_2$ of $E$. If $E$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{ε-c_1}{2}$ and $α-2$. The paper considers also the case of a threefold $X$ with $Pic(X) \ne \ZZ$ but $Num(X) \cong \ZZ$ and gives similar non-vanishing results.
The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle E on P^3 which force E to split, and finds therefore strategic levels of non-vanishing for a non-split bundle. The present conditions improve other conditions known in the literature and are obtained with simple computations on the Euler characteristic function, avoiding the speciality lemma, Barth's restriction theorem, the discriminat property, and other heavy tools.