[Paper Review] Covered by Lines and Conic Connected Varieties

Simone Marchesi, Alex Massarenti+1 more|arXiv (Cornell University)|May 30, 2011

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a numerical criterion for a projective variety covered by lines to be connected by singular conics, proving that if the sum of the degrees of its defining equations satisfies ∑di ≤ (N + m)/2, then two general points lie on a singular conic. The key contribution is a sharp bound that generalizes prior results on conic-connectedness, with applications to Fano, QEL, and LQEL varieties.

ABSTRACT

We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak.

Motivation & Objective

• To establish a numerical condition ensuring that two general points on a variety covered by lines lie on a singular conic.
• To generalize and refine existing criteria for conic-connectedness, particularly those of Bonavero and Höring, by allowing singular and non-smooth varieties.
• To explore the interplay between conic-connectedness, covered-by-lines varieties, and Fano geometry using classical results from Ein and Zak.
• To investigate the sharpness of the proposed criterion through explicit examples, including cubic hypersurfaces in P⁴.
• To compute the number of singular conics through two general points under equality conditions in the criterion.

Proposed method

• Use of the variety of minimal rational tangents Lx at a point x to analyze the geometry of lines through x.
• Application of Zak’s Theorem on Tangencies and Ein’s classification of dual defective varieties to derive bounds on the dimension of Lx.
• Derivation of a numerical inequality involving the degrees di of defining equations Gi and the ambient dimension N, leading to ∑di ≤ (N + m)/2.
• Use of Bezout’s theorem and intersection theory to count singular conics when equality holds in the criterion.
• Construction of a system of equations forcing conics through two points x and y by dehomogenizing the Gi and analyzing vanishing conditions up to degree di.
• Comparison with existing results, particularly Bonavero and Höring’s work on smooth complete intersections, to show the current criterion is more general and sharp.

Experimental results

Research questions

• RQ1 Under what numerical condition on the degrees of defining equations does a variety covered by lines admit a singular conic through two general points?
• RQ2 Is the proposed inequality ∑di ≤ (N + m)/2 sharp, and what examples demonstrate its sharpness?
• RQ3 How does conic-connectedness relate to other classes such as QEL, LQEL, and prime Fano varieties?
• RQ4 What is the number of singular conics through two general points when equality holds in the criterion?
• RQ5 In what cases does conic-connectedness fail even if the variety is covered by lines and Fano?

Key findings

• The inequality ∑_{i=1}^m d_i ≤ (N + m)/2 is a sufficient condition for a variety X ⊂ ℙ^N covered by lines to be connected by singular conics.
• The criterion is sharp: equality is achieved in the case of a smooth cubic threefold in ℙ⁴, which is conic-connected but not connected by singular conics.
• When ∑_{i=1}^c d_i = (N + c)/2 for a smooth complete intersection, the number of singular conics through two general points is finite and equals ∏_{i=1}^c d_i!(d_i - 1)!.
• For a smooth quadric in ℙ³, exactly 2 singular conics connect two general points, consistent with the formula.
• For a complete intersection of two quadrics in ℙ⁶, the formula predicts 4 singular conics, matching the geometric intersection of loci in tangent spaces.
• The result extends beyond smooth complete intersections: it applies to singular varieties and ensures existence of singular conics even when smooth conics do not exist.