176 research outputs found

    Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic

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    We determine the possible intersection sizes of a Hermitian surface H\mathcal H with an irreducible quadric of PG(3,q2){\mathrm PG}(3,q^2) sharing at least a tangent plane at a common non-singular point when qq is even.Comment: 20 pages; extensively revised and corrected version. This paper extends the results of arXiv:1307.8386 to the case q eve

    Intersections of the Hermitian surface with irreducible quadrics in PG(3,q2)PG(3,q^2), qq odd

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    In PG(3,q2)PG(3,q^2), with qq odd, we determine the possible intersection sizes of a Hermitian surface H\mathcal{H} and an irreducible quadric Q\mathcal{Q} having the same tangent plane π\pi at a common point P∈Q∩HP\in{\mathcal Q}\cap{\mathcal H}.Comment: 14 pages; clarified the case q=

    Szemer\'edi--Trotter-type theorems in dimension 3

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    We estimate the number of incidences in a configuration of mm lines and nn points in dimension 3. The main term is mn1/3mn^{1/3} if we work over the real or complex numbers but mn2/5mn^{2/5} over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.Comment: This paper supersedes arXiv:1404.4613. Version2: references updated and small changes made. arXiv admin note: substantial text overlap with arXiv:1404.4613 Version 3: Many changes and a new section added on ruled surface

    Intersection sets, three-character multisets and associated codes

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    In this article we construct new minimal intersection sets in AG(r,q2){\mathrm{AG}}(r,q^2) sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in PG(r,q2){\mathrm{PG}}(r,q^2) with rr even and we also compute their weight distribution.Comment: 17 Pages; revised and corrected result

    On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in P3\mathbb{P}^3

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    It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in P3\mathbb{P}^3 does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.Comment: 26 page

    Stability of restrictions of cotangent bundles of irreducible Hermitian symmetric spaces of compact type

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    It is known that the cotangent bundle ΩY\Omega_Y of an irreducible Hermitian symmetric space YY of compact type is stable. Except for a few obvious exceptions, we show that if X⊂YX \subset Y is a complete intersection such that Pic(Y)→Pic(X)Pic(Y) \to Pic(X) is surjective, then the restriction ΩY∣X\Omega_{Y|X} is stable. We then address some cases where the Picard group increases by restriction.Comment: Results and exposition improve

    Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

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    We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives

    tt-Intersection sets in AG(r,q2)AG(r,q^2) and two-character multisets in PG(3,q2)PG(3,q^2)

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    In this article we construct new minimal intersection sets in AG(r,q2)AG(r,q^2) with respect to hyperplanes, of size q2r−1q^2r-1 and multiplicity tt, where t∈ q2r−3−q(3r−5)/2,q2r−3+q(3r−5)/2−q(3r−3)/2$,fort\in \ q^2r-3-q^(3r-5)/2, q^2r-3+q^(3r-5)/2-q^(3r-3)/2\$, for roddor odd or t \in \ q^2r-3-q^(3r-4)/2, q^2r-3-q^r-2\,for, for reven.Asabyproduct,foranyodd even. As a byproduct, for any odd qwegetanewfamilyoftwo−charactermultisetsin we get a new family of two-character multisets in PG(3,q^2).Theessentialideaistoinvestigatesomepoint−setsin. The essential idea is to investigate some point-sets in AG(r,q^2)$ satisfying the opposite of the algebraic conditions required in [1] for quasi--Hermitian varieties

    Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface

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    In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree dd and a non-degenerate Hermitian surface in \PP^3(\Fqt). The conjecture was proven to be true by Edoukou in the case when d=2d=2. In this paper, we prove that the conjecture is true for d=3d=3 and q≥8q \ge 8. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degenerate Hermitian surface. This classifications disproves one of the conjectures proposed by Edoukou, Ling and Xing
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