We introduce various families of irreducible homaloidal hypersurfaces in projective space P r , for all r ≥ 3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan-Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge. * This author thanks CNPq for support and the Departamento de Matemática at Recife for hospitality during the preparation of this paper.
The number of apparent double points of a smooth, irreducible projective variety X of dimension n in P 2n+1 is the number of secant lines to X passing through the general point of P 2n+1 . This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow us to treat a wider class of projective varieties. Licensed to Univ of British Columbia. Prepared on Thu Jul 2 06:01:03 EDT 2015 for download from IP 142.103.160.110. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Licensed to Univ of British Columbia. Prepared on Thu Jul 2 06:01:03 EDT 2015 for download from IP 142.103.160.110. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf VARIETIES WITH ONE APPARENT DOUBLE POINT 477 the case d = 9 and to conclude that the other cases 5 ≤ d ≤ 8 correspond to the aforementioned varieties.It should be noted that M. G. Violo, in her Ph.D. thesis [V], also studied the same classification problem. She tried to adapt, to higher dimensions, Severi's argument for surfaces. However, in order to do so, she had to make a rather unnatural assumption and her list of OADP-threefolds does not include the degree 8 scroll.Our approach could in principle be applied also in higher dimensions. However, the analysis becomes then much heavier, to the extent of being in fact discouraging. What it suggests is that it should be possible to bound the degree of an OADP-variety of dimension n by a function depending on n. In other words, it should still be the case that, in any dimension, one has finitely many families of OADP-varieties. The bound that our approach suggests for the degree is exponential in n, and this is one of the reasons why the classification, in this way, becomes rather unfeasible. It could happen, however, that the cases which are really possible are much more restricted; after all, even in dimension 3, the case d = 9, is a priori possible according to our approach, does not in fact exist.One more piece of information that we are able to give is that both an OADP-variety and its general hyperplane section have Kodaira dimension −∞ (see Proposition 4.6). This suggests an alternative approach to the classification which we briefly outline in Remark 4.11. It also relates to an interesting conjecture of Bronowski [Br] to the effect that a variety X ⊂ P 2n+1 of dimension n has one apparent double point if and only if the projection of X to P n from the projective tangent space to X at its general point is birational (see Remark 4.3). Here we prove that, in general, the degree of the aforementioned projection is bounded above by the number of apparent double points of X, which yields one implication of Bronowski's conjecture (see Proposition 4.1 and Corollary 4.2). In particular, we have the important consequence that OADP-varieties are rational.Finally, as an applicati...
Quadratic entry locus manifold of type $\delta$ $X\subset\mathbb P^N$ of dimension $n\geq 1$ are smooth projective varieties such that the locus described on $X$ by the points spanning secant lines passing through a general point of the secant variety $SX\subseteq\mathbb P^N$ is a smooth quadric hypersurface of dimension $\delta=2n+1-\dim(SX)$ equal to the secant defect of $X$. These manifolds appear widely and naturally among projective varieties having special geometric properties and/or extremal tangential behaviour. We prove that, letting $\delta=2r_X +1\geq 3$ or $\delta=2r_X+2$, then $2^{r_X}$ divides $n-\delta$. This is obtained by the study of the projective geometry of the Hilbert scheme $Y_x\subset \mathbb(T_x^*)$ of lines passing through a general point $x$ of $X$, allowing an inductive procedure. The Divisibility Property described above allows unitary and simple proofs of many results on $QEL$-manifolds such as the complete classification of those of type $\delta\geq n/2$, of Cremona transformation of type $(2,3)$, $(2,5)$. In particular we propose a new and very short proof of the fact that Severi varieties have dimension 2,4, 8 or 16 and also an almost self contained half page proof of their classification due to Zak.Comment: 16 pages; some misprints and imprecisions corrected; some references added; final version as appeared in Math. An
We study a particular class of rationally connected manifolds, X ⊂ P N , such that two general points x, x ′ ∈ X may be joined by a conic contained in X. We prove that these manifolds are Fano, with b2 2. Moreover, a precise classification is obtained for b2 = 2. Complete intersections of high dimension with respect to their multi-degree provide examples for the case b2 = 1. The proof of the classification result uses a general characterization of rationality, in terms of suitable covering families of rational curves.A Gaetano Scorza che un secolo fa aveva colto l'importanza delle varietà razionalmente connesse considerando la classe particolare di varietà conicamente connesse formata da quelle di ultima specie."Invece per le V4 di prima e terza specie arrivo a caratterizzarle tutte valendomi della teoria dei sistemi lineari sopra una varietà algebrica e, per le ultime, della circostanza che esse contengono un sistema ∞ 6 di coniche così che per ogni loro coppia di punti passa una e una sola conica. Inoltre la natura dei ragionamentiè tale da mostrare come i risultati ottenuti possano estendersi, almeno per la maggior parte, alle varietà (di prima e ultima specie) a un numero qualunque di dimensioni, .
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