On the tangent degree and the degree of the tangent variety of a projective variety

Technical Summary: On the Tangent Degree and the Degree of the Tangent Variety of a Projective Variety

Problem Statement
The paper investigates two primary invariants associated with a projective variety $X^n \subset \mathbb{P}^N$: the tangent degree $\tau(X)$ and the degree of the tangent variety $\deg(\text{Tan}(X))$.

• The tangent degree $\tau(X)$ is defined as the number of tangent spaces to $X$ at smooth points passing through a general point of the tangent variety $\text{Tan}(X)$. If $\dim(\text{Tan}(X)) < 2n$, $\tau(X)$ is zero.
• The study focuses on the relationship between these invariants, particularly in cases where $N \ge 2n$ and where the tangent variety does not coincide with the secant variety ($\text{Tan}(X) \subsetneq \text{Sec}(X)$).
• A central motivation is the generalization of elementary facts regarding plane curves (where $\tau(X) \ge 2$ for $\deg(X) \ge 2$) to higher dimensions, specifically addressing the non-existence of varieties with $\tau(X)=1$ in the case $N=2n$ and the classification of varieties where $\tau(X)=2$.

Methodology
The authors employ a synthesis of classical projective geometry, differential geometry, and modern algebraic geometry techniques:

1. Focal Loci and Developable Families: The paper utilizes the theory of foci for families of linear spaces. It analyzes the differential of the tautological morphism $q: TX \to \text{Tan}(X)$ to identify focal points. The concept of "developable" varieties (where the tangent space is constant along certain subvarieties) is central to the analysis of singularities and dimension counts.
2. Second Fundamental Form: The authors extensively use the second fundamental form $|II_{x,X}|$ of the variety at a general point $x$. They relate the kernel of the differential of the Gauss map to the base locus of the second fundamental form and the existence of asymptotic directions.
3. Scorza Maps and Entry Loci: For varieties in $\mathbb{P}^{2n}$, the Scorza map is used to establish birationality between the variety and its general tangent space when $\tau(X)=2$. The properties of entry loci (specifically QEL-varieties with quadratic entry loci) are leveraged to classify smooth varieties with $\tau(X)=2$.
4. Induction and Projection: To establish lower bounds for $\deg(\text{Tan}(X))$ when $\text{Tan}(X) \subsetneq \text{Sec}(X)$, the authors use induction on the codimension, analyzing the behavior of degrees under generic projections from points on the variety.
5. Classification via Linear Systems: The classification of surfaces with $\tau(X) \ge 2$ involves analyzing the Gauss defect $\varrho(X)$ and the structure of the singular locus of the tangent variety, leading to the identification of specific scroll structures (Roth varieties).

Key Contributions and Results

1. The Tangent Degree in $\mathbb{P}^{2n}$

• Non-existence of $\tau(X)=1$: The paper proves that for an irreducible variety $X^n \subset \mathbb{P}^{2n}$, $\tau(X) \neq 1$. If $\text{Tan}(X) = \mathbb{P}^{2n}$, then $\tau(X) \ge 2$.
• Classification of $\tau(X)=2$:
  • If $X^n \subset \mathbb{P}^{2n}$ is smooth and $\tau(X)=2$, then $X$ is a QEL-variety of type $\delta=1$.
  • Even Dimension: The unique smooth even-dimensional varieties with $\tau(X)=2$ are rational normal scrolls $S(1, \dots, 1, 2) \subset \mathbb{P}^{4m}$.
  • Odd Dimension: Smooth odd-dimensional varieties with $\tau(X)=2$ are either rational normal scrolls or specific odd-dimensional Fano manifolds (e.g., prime del Pezzo threefolds and Mukai fivefolds).
  • Singular Cases: The authors construct singular examples of arbitrary large dimension and degree with $\tau(X)=2$, including projections of OADP-varieties (varieties with one apparent double point) and Verra varieties.

2. Lower Bounds for $\deg(\text{Tan}(X))$

• The paper establishes a linear lower bound for the degree of the tangent variety when $\text{Tan}(X) \subsetneq \text{Sec}(X)$.
• Theorem: If $X^n \subset \mathbb{P}^N$ is non-degenerate and $\text{Tan}(X) \subsetneq \text{Sec}(X)$, then:
  $$\deg(\text{Tan}(X)) \ge 2(N - \dim(\text{Tan}(X)) + 1)$$
• This bound is shown to be sharp. The equality is attained by smooth rational normal scrolls and certain singular varieties constructed as cones over them.
• In contrast, if $\text{Tan}(X) = \text{Sec}(X)$, the degree bound is quadratic in the codimension (referencing [6]).

3. Classification of Surfaces with $\tau(X) \ge 2$

• For non-degenerate irreducible surfaces $X^2 \subset \mathbb{P}^N$ with $N \ge 5$ and $\tau(X) \ge 2$, the paper provides a complete classification based on the Gauss defect $\varrho(X)$:
  • Case $\varrho(X)=2$: $N=5$, and $X$ is projectively equivalent to the Veronese surface $\nu_2(\mathbb{P}^2)$. Here $\tau(X)=2$.
  • Case $\varrho(X)=3$: $X$ is contained in a 3-dimensional developable scroll in planes $Y^3 \subset \mathbb{P}^N$. The general plane of the scroll intersects $X$ in a curve of degree $b \ge 2$.
• Smooth Linearly Normal Surfaces: If $X$ is smooth and linearly normal, the only possibilities are:
  1. The Veronese surface in $\mathbb{P}^5$ ($\tau=2$).
  2. Roth Surfaces: $X \subset S(L, Z) \subset \mathbb{P}^N$, where $S(L, Z)$ is a cone over a smooth rational normal scroll $Z$ with vertex a line $L$. These surfaces have degree $d = b(N-2) + 1$ and tangent degree $\tau(X) = b^2$ for some integer $b \ge 2$.

Significance and Claims
The authors position this work as a unification of dispersed results in the theory of foci, developable varieties, and secant defects.

• Unification: The paper collects and clarifies the relationship between the tangent degree, the second fundamental form, and the geometry of the tangent variety, providing a coherent framework for studying varieties where the tangent map is not birational.
• Classification: It offers a definitive classification for smooth varieties with $\tau(X)=2$ in $\mathbb{P}^{2n}$ and for surfaces with $\tau(X) \ge 2$ in higher codimensions.
• Geometric Characterization: The results provide a new geometric characterization of Roth varieties (smooth linearly normal varieties where the double divisor of a general projection is not ample) specifically for surfaces, linking them to the condition $\tau(X) \ge 2$ and $\text{Tan}(X) \subsetneq \text{Sec}(X)$.
• Sharpness: The paper demonstrates that the derived linear lower bound for the degree of the tangent variety is optimal, contrasting it with the quadratic bound for secant varieties.

The authors acknowledge that while the results for $n=2$ are complete, the existence of smooth QEL-manifolds of type $\delta=1$ with $\tau(X)=2$ for $n \ge 3$ (specifically odd dimensions $n=2m+1$ with $m \ge 3$) remains an open question, suggesting that the classification in Corollary 15 might not be exhaustive for higher dimensions.