[Paper Review] On the classification of defective threefolds

Luca Chiantini, Ciro Ciliberto|ArXiv.org|Dec 31, 2003

Algebraic Geometry and Number Theory21 references5 citations

TL;DR

This paper provides a complete classification of irreducible, non-degenerate, projective threefolds that are $k$-defective for $k \geq 2$, meaning their $k$-secant variety has dimension less than the expected value. Using tangential projections and Castelnuovo’s theory on Hilbert functions, the authors identify all minimally $k$-defective threefolds, showing they arise as $2$-uple embeddings or cones over such embeddings of threefolds or surfaces of minimal degree or specific degree and genus in $\mathbb{P}^{k+1}$, with precise conditions on dimension, defect, and embedding space.

ABSTRACT

We classify all irreducible projective threefolds $X$ which are $k$-defective, i.e. some $k$-secant variety of $X$ has dimension less than the expected value. This results extends the classical Scorza's classification of the case $k=1$.

Motivation & Objective

• To classify all irreducible, non-degenerate, projective threefolds that are $k$-defective for $k \geq 2$, i.e., whose $k$-secant variety has dimension less than expected.
• To extend the classification of defective varieties beyond the known cases of curves, surfaces, and $1$-defective threefolds.
• To focus on minimally $k$-defective threefolds, i.e., those not $h$-defective for any $h < k$, to provide a complete structural classification.
• To characterize such threefolds via geometric constructions, particularly $2$-uple embeddings and cones over varieties in $\mathbb{P}^{k+1}$ with specific degree and genus conditions.
• To unify and generalize previous results on defective varieties using tangential projections and invariants like $n_k(X)$ and $\delta_k(X)$.

Proposed method

• Employ tangential projections as the central geometric tool to analyze the structure of $k$-secant varieties and their contact loci.
• Use Castelnuovo’s theory on the growth of Hilbert functions to control the dimension of secant varieties and derive constraints on the varieties.
• Analyze the tangential contact locus—defined as the set of points where the tangent space to the secant variety agrees with that of the variety—classifying cases by its irreducibility and dimension.
• Distinguish three main cases: contact locus an irreducible divisor, reducible, or an irreducible curve, each leading to a different class of threefolds.
• Apply the invariant $n_k(X)$, measuring the dimension of the contact locus, to classify varieties according to their defectivity type.
• Use the fact that $k$-defective threefolds arise as $2$-uple embeddings or cones over such embeddings of threefolds or surfaces in $\mathbb{P}^{k+1}$ with prescribed degree and genus.

Experimental results

Research questions

• RQ1 What are all possible irreducible, non-degenerate, projective threefolds that are $k$-defective for $k \geq 2$?
• RQ2 How can the structure of the $k$-secant variety's contact locus determine the geometric type of a $k$-defective threefold?
• RQ3 What are the precise conditions on degree, genus, and embedding dimension that classify minimally $k$-defective threefolds?
• RQ4 How do constructions via $2$-uple embeddings and cones over varieties in $\mathbb{P}^{k+1}$ account for all such $k$-defective threefolds?
• RQ5 What role do the invariants $n_k(X)$ and $\delta_k(X)$ play in distinguishing the different classes of $k$-defective threefolds?

Key findings

• All minimally $k$-defective threefolds for $k \geq 2$ are classified into 15 distinct geometric types, with $r = 4k+2$ or $r = 4k+3$ as the ambient projective space dimension.
• For $k \geq 2$, $k$-defective threefolds are either $2$-uple embeddings of threefolds of degree $k-1$, $k$, or $k+1$ in $\mathbb{P}^{k+1}$, or cones over such embeddings with vertex of dimension $0$, $1$, $k-i$, or $k-1$.
• When the contact locus is an irreducible divisor, the threefold is either a $2$-uple embedding of a threefold of minimal degree $k-1$ in $\mathbb{P}^{k+1}$, or a cone over such a variety with vertex a point or line.
• When the contact locus is a curve, the threefold is a $2$-uple embedding of a threefold of degree $k$ in $\mathbb{P}^{k+1}$ with curve sections of arithmetic genus $1$ or $0$, or a cone over such a variety.
• For $k \geq 4$, there exist $k$-defective threefolds that are $2$-uple embeddings of threefolds of degree $k+1$ in $\mathbb{P}^{k+1}$ with curve sections of arithmetic genus $2$, embedded in $\mathbb{P}^{4k+3}$.
• The threefold $X$ obtained as the $2$-uple embedding of $\mathbb{P}^1 \times \mathbb{P}^2$ with the linear system $|\mathcal{O}(1,3)|$ is $4$-defective and fits into the classification as a cone over a surface of degree $k=4$ in $\mathbb{P}^5$, with $r=19$, $\delta_4(X)=1$, and $n_4(X)=2$.