Varieties with many lines

Rogora, Enrico

doi:10.1007/bf02567698

Cited by 35 publications

(39 citation statements)

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“…By a theorem of E. Rogora see [3] we know that in case n ≥ 4 and dim( ) = 2n − 4, then one of the following possibilities hold. (a) X contains a linear (n − 1)-space in P N ; (b) X is the union of a 1-dimensional family of quadric hypersurfaces (this means, there is a curve in Gr(n; N) and for each t ∈ there is a quadric hypersurface Q t in the associated linear n-space such that X is the union of those Q t ).…”

Section: Preliminariesmentioning

confidence: 96%

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Preservation of the injectivity of the Gauss map for general projections

Coppens

2007

Geom Dedicata

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Let X be a smooth irreducible quasi-projective variety of dimension n in P N with N ≥ 2n + 2. Let γ be its Gauss map, let X ⊂P N−1 be the embedding obtained from the general projection in P N and let γ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ (Q) = γ (Q ) implies γ (Q) = γ (Q ). We prove that this property holds in the following cases: N ≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n − 1 and X does contain a linear (n − 1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map.

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Section: Preliminariesmentioning

confidence: 96%

“…At some step we reduce the problem to the study of certain varieties X containing a family of lines of dimension 2n − 4. Then, in (2.2.4) we use the classification in [3] of such varieties.…”

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confidence: 99%

Preservation of the injectivity of the Gauss map for general projections

Coppens

2007

Geom Dedicata

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“…The following theorem is a major ingredient in the present part of our analysis. It has been obtained by Severi [34] in 1901, and a variant of it is also attributed to Segre [33]; it is mentioned in a recent work of Rogora [31], in another work of Mezzetti and Portelli [26], and also appears in the unpublished thesis of Richelson [30]. Severi's paper is not easily accessible (and is written in Italian).…”

Section: Since For Everymentioning

confidence: 99%

“…Note also that the cases where dim Σ 0 < 2k − 3 are not treated by the theorem (although they might occur); see Rogora [31] for a (partial) treatment of these cases.…”

Section: Since For Everymentioning

confidence: 99%

Incidences between points and lines in R4

Sharir

Solomon

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We show that the number of incidences between m distinct points and n distinct lines in R 4 is O 2 c √ log m (m 2/5 n 4/5 + m) + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor 2 c √ log m when m ≤ n 6/7or m ≥ n 5/3 . Except for the factor 2 c √ log m , the bound is tight in the worst case.Keywords. Combinatorial geometry, incidences, the polynomial method, algebraic geometry, ruled surfaces. IntroductionLet P be a set of m distinct points in R 4 and let L be a set of n distinct lines in R 4 . Let I(P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p, ℓ) with p ∈ P , ℓ ∈ L, and p ∈ ℓ. If all the points of P and all the lines of L lie in a common plane, then the classical Szemerédi-Trotter theorem [42] yields the worst-case tight boundThis bound clearly also holds in R 4 (or in any other dimension), by projecting the given lines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by placing all the points and lines in a common plane, in a configuration that yields the planar lower bound.In the recent groundbreaking paper of Guth and Katz [15], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3 , provided that not too many lines of L lie in a common plane 1 . Specifically, they showed: Theorem 1.1 (Guth and Katz [15]). Let P be a set of m distinct points and L a set of n distinct lines in R 3 , and let s ≤ n be a parameter, such that no plane contains more than s lines of L. ThenThis bound is tight in the worst case.In this paper, we establish the following analogous and sharper result in four dimensions.1 The additional requirement in [15], that no regulus contains too many lines, is not needed for the incidence bound given below.1 Theorem 1.2. Let P be a set of m distinct points and L a set of n distinct lines in R 4 , and let q, s ≤ n be parameters, such that (i) each hyperplane or quadric contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Thenwhere A and c are suitable absolute constants. When m ≤ n 6/7 or m ≥ n 5/3 , we get the sharper boundIn general, except for the factor 2 c √ log m , the bound is tight in the worst case, for any values of m, n, and for corresponding suitable ranges of q and s.The proof of Theorem 1.2 will be by induction on m. To facilitate the inductive process, we extend the theorem as follows. We say that a hyperplane or a quadric H in R 4 is q-restricted for a set of lines L and for an integer parameter q, if there exists a polynomial g H of degree at most O( √ q), such that each of the lines of L that is contained in H, except for at most q lines, is contained in some irreducible component of H ∩ Z(g H ) that is ruled by lines and is not a 2-flat (see below for details). In other words, a q-restricted hyperplane o...

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“…It is immediate to check that the union of the lines of the congruence has dimension exactly three. Thus the only main ingredient is a theorem of B. Segre (see [22], or [21] for a modern proof) stating that a threefold with a three-dimensional family of lines is either a linear space of dimension three, or a three-dimensional quadric, or it contains a one-dimensional family of planes (and the given family of lines of the threefold consists of those contained in these planes). Clearly the last two cases correspond to cases (ii) and (iii) in the statement (we can assume the quadric to be smooth, since otherwise it would contain a one-dimensional family of planes and we reduce to case (iii)).…”

Section: Order Zeromentioning

confidence: 99%

Line Congruences of Low Order

Arrondo

2002

Milan Journal of Mathematics

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Abstract. We present a survey on congruences of lines of low order. We start by recalling the main properties of the focal locus, and use it to reobtain the classification of congruences of order one in P 3 . We then explain the main ideas of a work in progress with S. Verra, outlining how to classify congruences of lines of order two in P 3 . We end by stating the main problems on this topic.The study of line congruences (i.e. (n − 1)-dimensional families of lines in P n ), especially for n = 3, 4 has been very popular in the turn from nineteenth till twentieth century, and has been retaken with modern techniques at the end of the last century. One of the aspects studied has been the classification of congruences of low order (the order is defined to be the number of lines of the family passing through a general point of P n ). In particular, line congruences in P 3 of order up to three have been studied among many others by Kummer ([15]) and Fano ([9] and [10]), while congruences of order one in P 4 have been studied especially by Marletta ([17] and [18]). Most of the results obtained by the classical geometers have been reobtained and improved in nowadays terms. Specifically, Ran ([20]) has given a complete classification of all the two-dimensional families of linear spaces of any dimension in any projective space and having order one (in particular, he reobtained, fixing one missing case, Kummer's classification of line congruences of order one in P 3 ). The cases of order two and three (still in P 3 ) were considered with modern techniques only when the family of lines was smooth. The classification of order two was obtained by Verra ([23]), while the classification of order three was completed by Gross ([14]). For line congruences in P 4 , the situation is much more complicated, and in fact only partial classifications are known for order one, even in case 224 E. Arrondo Vol. 70 (2002) the congruences are assumed to be smooth (see [7] and [8] for a thorough study of some of the cases outlined by Marletta).In this paper we will give an overview of the general situation, explaining how to obtain much of the above classification, and enlarging it to the case of line congruences of order two in P 3 without the smoothness hypothesis (the latter being a work in progress with Sandro Verra). The concrete structure and contents of the paper are as follows.In Section 1 we will fix the notation and basic definitions, in particular the one of the focal locus, a classical topic recently retaken by Ciliberto and Sernesi ([5]). We will then state and briefly prove the main properties of the focal locus of line congruences that we will need. A complete study of these (and many other) properties in the case of line congruences in P 3 (the main case treated in this paper) can be found in [3], while for an arbitrary P n the results will appear in a forthcoming paper of Marina Bertolini, Cristina Turrini and the author. Anyway most of the results of this section (and the next one) are very classical and has been reproved ...

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Varieties with many lines

Cited by 35 publications

References 7 publications

Preservation of the injectivity of the Gauss map for general projections

Preservation of the injectivity of the Gauss map for general projections

Incidences between points and lines in R4

Line Congruences of Low Order

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