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Regular and rigid curves on some Calabi-Yau and general-type complete intersections

Ziv Ran

Abstract

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi-Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n + 2$ in $\mathbb P^n$.

Regular and rigid curves on some Calabi-Yau and general-type complete intersections

Abstract

Let be either a general hypersurface of degree in or a general complete intersection in . We construct balanced rational curves on of all high enough degrees. If or , we construct rigid curves of genus on of all high enough degrees. As an application we construct some rigid bundles on Calabi-Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree in .
Paper Structure (4 sections, 7 theorems, 50 equations)

This paper contains 4 sections, 7 theorems, 50 equations.

Table of Contents

  1. Fan, fang and quasi-cone degenerations
  2. Relatively regular bundles and curves
  3. Lines and conics on some hypersurfaces
  4. Conclusions

Key Result

Theorem 1

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$, and let $e$ be an integer. Then (i) if $e\geq 2n-1$, there exist smooth rational curves of degree $e$ on $X$ with normal bundle $(n-4)\mathcal{O}\oplus2 If either $g=1$ or $n=4$, then there exists a smooth rigid curve of genus $g$ and degree $e$ on $X$

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Conjecture
  • Definition 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 13 more