Behavior of rational curves of the minimal degree in projective space bundle in any characteristic

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

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On 3-linear varieties of codimension 2

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501066 ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050106

Author(s):

Wanseok Lee ◽

Euisung Park

Keyword(s):

Projective Space ◽

Projective Variety ◽

Minimal Degree

A projective variety in a projective space is said to be [Formula: see text]-linear if it is [Formula: see text]-regular and has no defining equation of degree [Formula: see text]. It is well known that [Formula: see text]-linear varieties are exactly varieties of minimal degree. In this paper, we study [Formula: see text]-linear varieties of codimension [Formula: see text]. We classify all smooth [Formula: see text]-linear varieties of codimension 2. There are six kinds of such varieties. Also, we provide some nonconic singular [Formula: see text]-linear varieties of codimension [Formula: see text].

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The Grothendieck group of a quantum projective space bundle

K-Theory ◽

10.1007/s10977-006-0020-5 ◽

2006 ◽

Vol 37 (3) ◽

pp. 263-289 ◽

Cited By ~ 3

Author(s):

Izuru Mori ◽

S. Paul. Smith

Keyword(s):

Projective Space ◽

Grothendieck Group ◽

Projective Space Bundle ◽

Quantum Projective Space

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COMPLETE INTERSECTIONS AND MOVABLE CURVES ON THE MODULI SPACE OF SIX-POINTED RATIONAL CURVES

International Journal of Algebra and Computation ◽

10.1142/s021819671250049x ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250049

Author(s):

PAUL L. LARSEN

Keyword(s):

Projective Space ◽

Complete Intersection ◽

Moduli Space ◽

Complex Projective Space ◽

Projective Variety ◽

Rational Curves ◽

Complete Intersections ◽

Family Of Curves ◽

Complex Projective

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves and the permutohedral or Losev–Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of this family, but differ for the non-toric member, the moduli space of six-pointed rational curves. The proof is via an algorithm that applies in greater generality. We also give an example of a projective toric threefold for which these two cones differ.

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Basepoint Free Cycles on $\overline{\operatorname{\textbf{M}}}_{\boldsymbol{0,n}}$ from Gromov–Witten Theory

International Mathematics Research Notices ◽

10.1093/imrn/rnz184 ◽

2019 ◽

Author(s):

P Belkale ◽

A Gibney

Keyword(s):

Projective Space ◽

Moduli Space ◽

Homogeneous Spaces ◽

Type A ◽

Rational Curves ◽

Conformal Blocks ◽

Witten Theory ◽

Level 1

Abstract Basepoint free cycles on the moduli space $\overline{\operatorname{M}}_{0,n}$ of stable $n$-pointed rational curves, defined using Gromov–Witten invariants of smooth projective homogeneous spaces are introduced and studied. Intersection formulas to find classes are given. Gromov–Witten divisors for projective space are shown equivalent to conformal blocks divisors for type A at level 1.

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On the irreducibility of Hilbert scheme of surfaces of minimal degree

Open Mathematics ◽

10.2478/s11533-012-0130-7 ◽

2013 ◽

Vol 11 (2) ◽

Cited By ~ 2

Author(s):

Fedor Bogomolov ◽

Viktor Kulikov

Keyword(s):

Projective Plane ◽

Hilbert Scheme ◽

Special Class ◽

Main Idea ◽

Rational Curves ◽

Minimal Degree ◽

Irreducible Components

AbstractThe article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

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