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CICY3 Families of Nonsingular Codimension two K3

Geoffrey Mboya

TL;DR

This work classifies codimension-two CICY3 fibred by K3 surfaces $S_{2,3}\subset\mathbb{P}^4$ inside rank-2 toric ambient spaces. It parameterizes by $(p,a_2,\ldots,a_5)$ with $0=a_1\le a_2\le a_3\le a_4\le a_5$, proves smoothness under basepoint-freeness and identifies isolated singularities via base loci, and enumerates 12 explicit $(p,a_2,\ldots,a_5)$ families (with some yielding nonsingular $X$ and others featuring a controlled set of singular points). For the first seven families, cohomology is computed via the Cayley trick and Jacobian rings to determine $h^{2,1}(X)$, with a toric Lefschetz analogue ensuring $H^{1,1}(X)=H^{1,1}(\mathbb{F}_A)$; a Macaulay2 implementation is described and the authors outline extending to 83 more families. The results enrich the catalog of CICY3 with K3-fibration structures and provide data useful for applications in SCFT constructions and further studies of singularities in fibered Calabi–Yau threefolds.

Abstract

We conduct a systematic search of codimension 2 Complete Intersection Calabi--Yau threefolds (CICY3) in rank 2 toric ambient spaces and fibered by complete intersection of a quadric and a cubic in $\C¶^4$. We classify both the nonsingular ones as well as those with isolated singularities.

CICY3 Families of Nonsingular Codimension two K3

TL;DR

This work classifies codimension-two CICY3 fibred by K3 surfaces inside rank-2 toric ambient spaces. It parameterizes by with , proves smoothness under basepoint-freeness and identifies isolated singularities via base loci, and enumerates 12 explicit families (with some yielding nonsingular and others featuring a controlled set of singular points). For the first seven families, cohomology is computed via the Cayley trick and Jacobian rings to determine , with a toric Lefschetz analogue ensuring ; a Macaulay2 implementation is described and the authors outline extending to 83 more families. The results enrich the catalog of CICY3 with K3-fibration structures and provide data useful for applications in SCFT constructions and further studies of singularities in fibered Calabi–Yau threefolds.

Abstract

We conduct a systematic search of codimension 2 Complete Intersection Calabi--Yau threefolds (CICY3) in rank 2 toric ambient spaces and fibered by complete intersection of a quadric and a cubic in . We classify both the nonsingular ones as well as those with isolated singularities.
Paper Structure (4 sections, 6 theorems, 66 equations, 2 tables)

This paper contains 4 sections, 6 theorems, 66 equations, 2 tables.

Table of Contents

  1. Preliminaries
  2. Construction: Threefold families of K3 $X_{2,3}\subset\mathbb{P}^4$ in $\mathbb{F}_A$
  3. Proof of Theorem \ref{['CYttci']}
  4. Cohomology of Codimension Two CICY3 in $\mathbb{F}_A$

Key Result

Theorem 2

Let $X=\mathbb{V}(f_1,f_2),$ embedded in $\mathbb{F}_A=\mathbb{F}(0,a_2,a_3,a_4,a_5),$ be a threefold fibred over $\mathbb{P}^1$ by complete intersections $S_{2,3}\subset\mathbb{C}\mathbb{P}^4$ of a quadric and a cubic. Let $D_1, D_2\in\operatorname{Pic}(\mathbb{F}_A)$ with $D_1+D_2=-K_{\mathbb{F}_A The resulting threefold $X$ is Calabi--Yau with at most isolated singularities in 12 cases along ei

Theorems & Definitions (13)

  • Definition 1: Fivefold Smooth Scroll
  • Theorem 2
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Theorem 3.2: Lefschetz Hyperplane Theorem VoisinHodge
  • Remark 3.3
  • Definition 3.4: Rank 3 Toric variety
  • Proposition 3.5
  • Proposition 3.6
  • ...and 3 more