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Projective space
About: Projective space is a research topic. Over the lifetime, 8014 publications have been published within this topic receiving 118482 citations.
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24 Jan 1980
TL;DR: The first properties of the plane can be found in this article, where the authors define the following properties: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. Ovals 9. Arithmetic of arcs of degree two 10. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes 15.
Abstract: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References
1,593 citations
TL;DR: In this article, the authors present a legal opinion on the applicability of commercial or impression systématiques in the context of the agreement of publication mathématique de l'I.H.S.
Abstract: © Publications mathématiques de l’I.H.É.S., 1994, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
1,192 citations
TL;DR: In this paper, the authors prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories, where the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves.
Abstract: We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.
808 citations
TL;DR: In this article, it was shown that there is always at least one collineation of period q with respect to any point in the projective plane PG(2, pn) for every prime p and positive integer n.
Abstract: A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let
794 citations