[Paper Review] On quadrisecant lines of threefolds in P^5

Emilia Mezzetti|ArXiv.org|Dec 7, 2001

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper classifies smooth threefolds in $\mathbb{P}^5$ whose quadrisecant lines do not fill the ambient space, proving that such threefolds either lie on a cubic hypersurface or contain a two-dimensional family of plane curves of degree at least four. The analysis extends Severi’s classical approach to surfaces by studying Fano schemes of lines on hyperplane sections and applying results on line-covered threefolds in $\mathbb{P}^4$, leading to a complete classification under non-degeneracy and non-quadric assumptions.

ABSTRACT

We study smooth threefolds of the projective space of dimension 5 whose quadrisecant lines don't fill up the space. We give a complete classification of those threefolds X whose only quadrisecant lines are the lines contained in X. Then we prove that, if X admits "true" quadrisecant lines, but they don't fill up the space, then either X is contained in a cubic hypersurface, or it contains a family of dimension at least two of plane curves of degree at least four.

Motivation & Objective

• To classify smooth threefolds in $\mathbb{P}^5$ whose quadrisecant lines do not fill the space.
• To extend Severi’s classification of surfaces without apparent triple points to the threefold case using Fano schemes of lines.
• To determine the geometric structure of threefolds admitting 'true' quadrisecant lines that do not span $\mathbb{P}^5$, under non-degeneracy and non-quadric assumptions.
• To analyze the Fano scheme of lines on hyperplane sections of such threefolds and relate their geometry to the global structure of the threefold.

Proposed method

• Analyzes the Fano scheme $\Sigma$ of lines on hyperplane sections $V \subset \mathbb{P}^4$ of the threefold's secant hypersurface $Y \subset \mathbb{P}^5$, focusing on dimension and reducedness.
• Applies Theorem 0.1 from [12] to classify threefolds $V \subset \mathbb{P}^4$ covered by a 2-dimensional family of lines, based on the number $\mu$ of lines through a general point.
• Uses the condition of generic reducedness of the Fano scheme to rule out fixed tangent planes and classify non-reduced components as cones or tangent-line unions.
• Applies monoidal constructions and projection arguments to rule out low-degree threefolds and show that projections from points or lines lead to contradictions unless $X$ lies on a cubic or has special curve families.
• Performs case-by-case analysis of the four cases in Theorem 0.1, excluding projections of complete intersections and Grassmannians due to degree and smoothness constraints.
• Employs Castelnuovo’s bound to show that if $X$ is $k$-secant to a quadric cone, then $X$ must be contained in a quadric, leading to contradiction under the non-quadric assumption.

Experimental results

Research questions

• RQ1 What is the complete classification of smooth threefolds in $\mathbb{P}^5$ whose only quadrisecant lines are those contained in the threefold itself?
• RQ2 Under what conditions do 'true' quadrisecant lines (not contained in the threefold) fail to fill $\mathbb{P}^5$, and what geometric structure must the threefold possess in such cases?
• RQ3 How does the structure of the Fano scheme of lines on hyperplane sections of a threefold relate to the global geometry of the threefold in $\mathbb{P}^5$?
• RQ4 Can a threefold with a 4-dimensional family of quadrisecant lines avoid lying on a cubic hypersurface or containing a 2-dimensional family of plane curves of degree $\geq 4$?
• RQ5 What constraints do degree and sectional genus impose on threefolds whose quadrisecant lines do not span $\mathbb{P}^5$?

Key findings

• If a smooth threefold $X \subset \mathbb{P}^5$ has no quadrisecant lines other than those contained in $X$, then $X$ is either a rational normal scroll or lies on a cubic hypersurface.
• If $X$ admits a 5-dimensional family of quadrisecant lines but they do not fill $\mathbb{P}^5$, then $X$ is either contained in a cubic hypersurface or contains a 2-dimensional family of plane curves of degree at least four.
• For a threefold $X$ not contained in a quadric, any 4-dimensional family of quadrisecant lines that do not fill $\mathbb{P}^5$ implies that the secant hypersurface $Y$ is either a cubic or contains a 2-dimensional family of planes cutting $X$ in curves of degree $k \geq 4$.
• The analysis rules out the possibility of $X$ being a projection of a complete intersection of two quadrics in $\mathbb{P}^6$, a section of $G(1,4)$, or a hyperplane section of $\mathbb{P}^2 \times \mathbb{P}^2$, due to degree and smoothness constraints.
• The case of $Y$ birationally fibered by smooth quadric surfaces leads to a contradiction, as it implies $X$ must be contained in a quadric, violating the non-quadric assumption.
• The only consistent geometric configurations are: $X$ contained in a cubic hypersurface, or $X$ containing a 2-dimensional family of plane curves of degree $k \geq 4$.