Quintic threefolds with triple points

TL;DR

This paper investigates the geometric properties of quintic threefolds with only triple point singularities, establishing bounds on their singularities and providing explicit examples.

Contribution

It proves an upper bound of 10 on the number of triple points for certain reducible hyperplane sections and constructs examples illustrating this bound.

Findings

Maximum of 10 triple points for reducible hyperplane sections

Construction of examples with various numbers of triple points

Discussion of the defect of these threefolds

Abstract

We study the geometry of quintic threefolds $X\subset \mathbb{P}^4$ with only ordinary triple points as singularities. In particular, we show that if a quintic threefold $X$ has a reducible hyperplane section then $X$ has at most $10$ ordinary triple points, and that this bound is sharp. We construct various examples of quintic threefolds with triple points and discuss their defect.