On threefolds covered by lines

Mezzetti, Emilia; Portelli, Dario

doi:10.1007/bf02940914

Cited by 11 publications

(12 citation statements)

References 10 publications

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“…Moreover, there are three lines passing through each point of this submanifold. Algebraic threefolds covered by lines were recently classified by Mezzetti and Portelli in [18]. It was demonstrated that threefolds with three lines through each point are intersections of G(1, 4) with a P 6 , that is, Plücker images of linear congruences.…”

Section: Examplementioning

confidence: 99%

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

Agafonov¹,

Ferapontov²

2001

manuscripta mathematica

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We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in P 4 such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronesé variety V 2 ⊂ P 5 and the theory of associativity equations of two-dimensional topological field theory.

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Section: Examplementioning

confidence: 99%

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

Agafonov¹,

Ferapontov²

2001

manuscripta mathematica

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“…. , D k , all upper bounded by the degree D chosen in (26) for the original values of m and n. The points of P have been partitioned among the zero sets Z(f 1 ), . .…”

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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“…One of my favorite problems to put on an undergraduate differential geometry exam is: Prove that a surface in Euclidean three space that has more than two lines passing through each point is a plane (i.e., has an infinite number of lines passing through). In [30], Mezzetti and Portelli showed that a 3-fold having more than six lines passing through a general point must have an infinite number. Show that an n-fold having more than n!…”

Section: Bertini Type Theorems and Applicationsmentioning

confidence: 99%

Differential Geometry of Submanifolds of Projective Space

Landsberg

2008

Symmetries and Overdetermined Systems of Partial Differential Equations

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Abstract. These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. The results of [16,20,29,18,19,10,31] are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem. Overview• Introduction to the local differential geometry of submanifolds of projective space.• Introduction to moving frames for projective geometry.• How much must a submanifold X ⊂ P N resemble a given submanifold Z ⊂ P M infinitesimally before we can conclude X ≃ Z? • To what order must a line field on a submanifold X ⊂ P N have contact with X before we can conclude the lines are contained in X? • Applications to algebraic geometry.• A new variant of the Hwang-Yamaguchi rigidity theorem.• An exposition of the Hwang-Yamaguchi rigidity theorem in the language of moving frames. Representation theory and algebraic geometry are natural tools for studying submanifolds of projective space. Recently there has also been progress the other way, using projective differential geometry to prove results in algebraic geometry and representation theory. These talks will focus on the basics of submanifolds of projective space, and give a few applications to algebraic geometry. For further applications to algebraic geometry the reader is invited to consult chapter 3 of [11] and the references therein.Due to constraints of time and space, applications to representation theory will not be given here, but the interested reader can consult [23] for an overview. Entertaining applications include new proofs of the classification of compact Hermitian symmetric spaces, and of complex simple Lie algebras, based on the geometry of rational homogeneous varieties (instead of root systems), see [22]. The applications are not limited to classical representation theory. There are applications to Deligne's conjectured categorical generalization of the exceptional series [25], to Vogel's proposed Universal Lie algebra [27], and to the study of the intermediate Lie algebra Notations, conventions. I mostly work over the complex numbers in the complex analytic category, although most of the results are valid in the C ∞ category and over other fields, even characteristic p, as long as the usual precautions are taken. When working over R, some results become more complicated as there are more possible normal forms. I use notations and the ordering of roots as in [2] and label maximal parabolic subgroups accordingly, e.g., P k refers to the maximal parabolic obtained by omitting the spaces corresponding to the simple root α k . v 1 , ..., v k denotes the linear span of the vectors v 1 , ..., v k . If X ⊂ PV is a subset,X ⊂ V denotes the corresponding cone in V \0, the inverse image of X under the projection V \0 → PV . and X ⊂ PV denotes the Zariski closure of X, the zero set of all the homogeneous polynomials vanis...

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On threefolds covered by lines

Cited by 11 publications

References 10 publications

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

Systems of conservation laws of Temple class, equations of associativity and linear¶congruences in P 4

Incidences between points and lines in R4

Differential Geometry of Submanifolds of Projective Space

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