Number of triple points on complete intersection Calabi-Yau threefolds

TL;DR

This paper establishes bounds on the maximum number of ordinary triple points on complete intersection Calabi-Yau threefolds in various projective spaces, providing specific examples and exact bounds for certain cases.

Contribution

It derives new bounds for triple points on Calabi-Yau threefolds and presents explicit examples achieving these bounds, advancing understanding of their singularities.

Findings

Maximum of 10 triple points in P5 for certain intersections

Exact bound of 10 triple points on a sextic hypersurface in P[1:1:1:1:2]

Identification of Calabi-Yau threefolds that cannot have triple points

Abstract

We discuss bounds for the number of ordinary triple points on complete intersection Calabi-Yau threefolds in projective spaces and for Calabi-Yau threefolds in weighted projective spaces. In particular, we show that in P5 the intersection of a quadric and a quartic cannot have more than 10 ordinary triple points. We provide examples of complete intersection Calabi-Yau threefolds with multiple triple points. We obtain the exact bound for a sextic hypersurface in P[1 : 1 : 1 : 1 : 2], which is 10. We also discuss Calabi-Yau threefolds that cannot admit triple points.