scholarly journals GV-spectroscopy for F-theory on genus-one fibrations

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Paul-Konstantin Oehlmann ◽  
Thorsten Schimannek
Keyword(s):  
Complete Intersection ◽  
Complete Intersections ◽  
Detailed Knowledge ◽  
Gauge Groups ◽  
Frobenius Method ◽  
Codimension Two ◽  
Novel Technique ◽  
Genus One

Abstract We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = ℤ4 that is engineered by fibrations of bi-quadrics in ℙ3. We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

Birational Mori fiber structures of ℚ-Fano 3-fold weighted complete intersections, II

2018 ◽  
Vol 2018 (738) ◽  
pp. 73-129 ◽  
Author(s):  
Takuzo Okada
Keyword(s):  
Complete Intersection ◽  
Complete Intersections ◽  
Fiber Space ◽  
Codimension Two ◽  
Fano Threefold

Abstract In [T. Okada, Birational Mori fiber structures of \mathbb{Q} -Fano 3-fold weighted complete intersection, Proc. Lond. Math. Soc. (3) 109 2014, 6, 1549–1600], we proved that, among 85 families of \mathbb{Q} -Fano threefold weighted complete intersections of codimension two, 19 families consist of birationally rigid varieties and the remaining families consists of birationally non-rigid varieties. The aim of this paper is to study systematically the remaining families and prove that every quasismooth member of 14 families is birational to another \mathbb{Q} -Fano threefold but not birational to any other Mori fiber space.

scholarly journals Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

2019 ◽  
Author(s):  
Sanghyeon Lee ◽  
Jeongseok Oh
Keyword(s):  
Algebraic Geometry ◽  
Vector Bundle ◽  
Complete Intersection ◽  
Comparison Theorem ◽  
Euler Number ◽  
Projective Spaces ◽  
Complete Intersections ◽  
Dimension 2 ◽  
Definition Of ◽  
Genus One

Abstract In [ 17, 18], Zinger defined reduced Gromov–Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). In [ 3], Chang and Li provided a proof of the comparison theorem for quintic Calabi–Yau three-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. In [ 5], Coates and Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger in [ 14] and proved the comparison theorem for every Calabi–Yau threefold. In this paper, we prove the comparison theorem for every (not necessarily Calabi–Yau) complete intersection of Dimension 2 or 3 in projective spaces by taking a definition of reduced GW invariants in [ 5].

scholarly journals Complete intersections primitive structures on space curves

2015 ◽  
Vol 26 (12) ◽  
pp. 1550104
Author(s):  
Philippe Ellia
Keyword(s):  
Complete Intersection ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Complete Intersections ◽  
Multiple Structure ◽  
Space Curves ◽  
Structure Formula ◽  
Primitive Structure

A multiple structure [Formula: see text] on a smooth curve [Formula: see text] is said to be primitive if [Formula: see text] is locally contained in a smooth surface. We give some numerical conditions for a curve [Formula: see text] to be a primitive set theoretical complete intersection (i.e. to have a primitive structure which is a complete intersection).

scholarly journals Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Svetlana Ermakova
Keyword(s):  
Vector Bundle ◽  
Complete Intersection ◽  
Finite Rank ◽  
Vector Bundles ◽  
Complete Intersections ◽  
Finite Codimension ◽  
Line Bundles ◽  
Direct Sum ◽  
Rank Vector

AbstractIn this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

scholarly journals Normal bundles of rational curves on complete intersections

2019 ◽  
Vol 21 (02) ◽  
pp. 1850011 ◽  
Author(s):  
Izzet Coskun ◽  
Eric Riedl
Keyword(s):  
Complete Intersection ◽  
Normal Bundle ◽  
Rational Curves ◽  
Complete Intersections ◽  
Type Formula ◽  
Normal Bundles ◽  
Rationally Connected

Let [Formula: see text] be a general Fano complete intersection of type [Formula: see text]. If at least one [Formula: see text] is greater than [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. If all [Formula: see text] are [Formula: see text] and [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that [Formula: see text] is separably rationally connected by exhibiting very free rational curves in [Formula: see text] of optimal degrees.

scholarly journals Recursive strategy for decomposing Betti tables of complete intersections

2019 ◽  
Vol 29 (07) ◽  
pp. 1165-1191
Author(s):  
Courtney R. Gibbons ◽  
Robert Huben ◽  
Branden Stone
Keyword(s):  
Detailed Analysis ◽  
Complete Intersection ◽  
Main Tool ◽  
Decomposition Algorithm ◽  
Complete Intersections ◽  
Recursive Decomposition ◽  
Alternative Algorithm ◽  
Size Condition ◽  
Recursive Decomposition Algorithm ◽  
Betti Diagram

In the spirit of Boij–Söderberg theory, we introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to investigate stability and compatibility of the Boij–Söderberg decompositions of related diagrams; indeed, when the biggest generating degree is sufficiently large, the alternative algorithm produces the Boij–Söderberg decomposition. We also provide a detailed analysis of the Boij–Söderberg decomposition for Betti diagrams of codimension four complete intersections where the largest generating degree satisfies the size condition.

scholarly journals Some properties of complete intersections in “good” projective varieties

1976 ◽  
Vol 61 ◽  
pp. 103-111 ◽  
Author(s):  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
Singular Surface ◽  
Complete Intersections ◽  
Closed Field ◽  
Irreducible Curve ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Torsion Free

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

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

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.

scholarly journals Spaces of rational curves on complete intersections

2013 ◽  
Vol 149 (6) ◽  
pp. 1041-1060 ◽  
Author(s):  
Roya Beheshti ◽  
N. Mohan Kumar
Keyword(s):  
Complete Intersection ◽  
Rational Curves ◽  
Complete Intersections ◽  
Constant Dimension ◽  
Low Degree

AbstractWe prove that the space of smooth rational curves of degree $e$ on a general complete intersection of multidegree $(d_1, \ldots , d_m)$ in $\mathbb {P}^n$ is irreducible of the expected dimension if $\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and $n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum _{i=1}^m d_i$ is small compared to $n$.

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