Inflectional loci of scrolls

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let \L=\mathcal O_{\mathbb P^N}(1)|_X denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of \L are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.Comment: 9 pages, improved version. Accepted in Mathematische Zeitschrif

Osculation for conic fibrations

Smooth projective surfaces fibered in conics over a smooth curve are investigated with respect to their k-th osculatory behavior. Due to the bound for the dimension of their osculating spaces they do not differ at all from a general surface for k=2, while their structure plays a significant role for k \geq 3. The dimension of the osculating space at any point is studied taking into account the possible existence of curves of low degree transverse to the fibers, and several examples are discussed to illustrate concretely the various situations arising in this analysis. As an application, a complete description of the osculatory behavior of Castelnuovo surfaces, i.e. rational surfaces whose hyperplane sections correspond to a linear system of nodal quartic plane curves. is given. The case k=3 for del Pezzo surfaces is also discussed, completing the analysis done for k=2$ i a previous paper of the authors (2001). Moreover, for conic fibrations X in P^N, whose k-th inflectional locus has the expected codimension a precise description of this locus is provided in terms of Chern classes. In particular, for N=8, it turns out that either X is hypo-osculating for k=3, or its third inflectional locus is 1-dimensional

Jets of antimulticanonical bundles on Del Pezzo surfaces of degree \leq 2

Let S be a complex Del Pezzo surface with K_S^2 \leq 2, and let r=4-K_S^2. The line bundle L=-rK_S being very ample, we investigate the k-jet spannedness of -tK_S for t \geq r. A key point is the stratification given by the rank of the evaluation map j_2:S \times H^0(S,L) \to J_2L, with values in the second jet bundle of L, which puts in evidence some relevant loci related to both the intrinsic and the extrinsic geometry of S. In particular, the generic 2-jet spannedness of L allows us to consider the second dual variety of (S,L), parameterizing the osculating hyperplanes to S \subset P^6, embedded by |L|. Its behavior turns out to be completely different in the two cases K_S^2=2 and K_S^2=1

Projective bundles enveloping rational conic fibrations and osculation

Motivated by previous research on the osculation for special varieties, we investigate rational conic fibrations in connection with their enveloping projective bundles with the aim of comparing their inflectional loci. Specific existence results for such pairs are established. As a consequence, the possible various numerical characters of the pairs involved for low values of the sectional genus are determined

Osculating properties of decomposable scrolls

Osculating spaces of decomposable scrolls (of any genus and not necessarily normal) are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus -deriving from flexes- are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls

Osculatory behavior and second dual varieties of Del Pezzo surfaces

Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

Jets of antimulticanonical bundles on Del Pezzo surfaces of degree (>=( 2

Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal