scholarly journals Landau–Ginzburg/Calabi–Yau Correspondence for a Complete Intersection via Matrix Factorizations

2021 ◽  
Author(s):  
Yizhen Zhao
Keyword(s):  
Complete Intersection ◽  
Quantum Cohomology ◽  
Projective Line ◽  
Matrix Factorizations ◽  
Complete Intersections ◽  
Cohomology Groups ◽  
Isolated Singularities ◽  
Quantum Invariants ◽  
Derived Equivalences

Abstract By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics.

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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