Smooth Projective Variety Overview

• Smooth projective varieties are defined as irreducible, closed subvarieties in projective space over ℂ whose affine cones are nonsingular outside the origin.
• Theorems A and B establish that numerical properties, such as degree divisibility and codimension bounds, govern the behavior of embedded subvarieties under specific dimension constraints.
• Quadratic varieties are further classified by their swept-out high-dimensional quadrics, merging classical geometric conditions with modern cohomological techniques.

A smooth projective variety is an irreducible, closed subvariety $X \subset \mathbb{P}^N$ over $\mathbb{C}$ that is smooth in the sense that its affine cone is smooth away from the origin; equivalently, the tangent space at every point has the expected dimension. These objects serve as fundamental building blocks in algebraic geometry, connecting topology, complex geometry, and projective algebraic geometry. In the context of embedded projective varieties, geometric and numerical constraints on their subvarieties reveal deep connections between intrinsic and extrinsic properties, especially for subvarieties of small codimension and varieties governed by quadratic equations (Li, 2012).

1. Definition and Structural Properties

Let $X \subset \mathbb{P}^N$ denote a smooth projective variety of complex dimension $n$. $X$ is projective if it is Zariski-closed in projective space, and smooth if the affine cone over $X$ is nonsingular outside the origin. $X$ is nondegenerate if it is not contained in any hyperplane of $\mathbb{P}^N$. The projective codimension is defined as $\operatorname{codim}_{\mathbb{P}^N} X = N-n$. The degree of $X$, $\deg(X)$, is the number of intersection points with a general linear subspace of codimension $n$; equivalently, $\deg(X) = \int_X H^n$ for $H$ the hyperplane class.

For any subvariety $Y \subset X \subset \mathbb{P}^N$, its linear span $\operatorname{span}(Y)$ is the smallest linear subspace containing $Y$. If $r = \dim(\operatorname{span}(Y))$, then the codimension of $Y$ in its span is $$\operatorname{codim}_{\operatorname{span}(Y)} Y = r - \dim(Y)$$.

2. Numerical and Geometric Constraints on Subvarieties

Smooth projective varieties and their subvarieties exhibit strong restrictions when the subvariety has "small codimension." Two central theorems articulate these restrictions:

Theorem A (Complete Intersection Case):

Let $X \subset \mathbb{P}^N$ be a nondegenerate smooth complete intersection of dimension $n$, and $Y \subset X$ an $m$-dimensional subvariety. If $m > n/2$, then:

1. $\deg(X)$ divides $\deg(Y)$;
2. $$\operatorname{codim}_{\operatorname{span}(Y)} Y \geq \operatorname{codim}_{\mathbb{P}^N} X$$.

Theorem B (General Case):

Let $X \subset \mathbb{P}^N$ be a nondegenerate smooth projective variety of dimension $n$, and $Y \subset X$ an $m$-dimensional subvariety. If $m \geq N/2$, then the same conclusions as Theorem A hold. The numeric bounds in both theorems are sharp (Li, 2012).

3. Cohomological Underpinnings and Proof Sketch

The divisibility condition $\deg(X) \mid \deg(Y)$ is demonstrated via either the Lefschetz theorem (complete intersection) or the Barth–Larsen theorem (general case), establishing isomorphisms $$H_{2n-2m}(\mathbb{P}^N, \mathbb{Z}) \rightarrow H_{2n-2m}(X, \mathbb{Z})$$ under the respective hypotheses. For a general linear subspace $L \subset \mathbb{P}^N$ of dimension $N-n+m$, $L \cap X$ has degree $\deg(X)$; any $m$-dimensional subvariety $Y$ corresponds to a multiple of the generator in $H_{2n-2m}(X)$, yielding $\deg(Y) = r \cdot \deg(X)$ for some integer $r$.

To establish the lower bound on $\operatorname{codim}_{\operatorname{span}(Y)} Y$, assume the contrary and consider a sequence of hyperplane sections through the span of $Y$ to produce a subvariety $X_c$ of dimension $>m$ still contained in the span. Nondegeneracy and the application of the Lefschetz theorem lead to a contradiction with the divisibility property, enforcing the geometric bound.

4. Sharpness and Illustrative Examples

The bounds in Theorems A and B are optimal. In the case of complete intersections of two quadrics in $\mathbb{P}^{n+2}$, the bound $m > n/2$ in Theorem A is tight: for even $n$ with $m = n/2$, the conclusions fail. When $N$ is odd, $X = G(1,4) \subset \mathbb{P}^9$ (the Plücker embedding, dimension 6) and $Y = G(1,H) \subset X$ of dimension 4 furnish $\deg(X)=5$, $\deg(Y)=2$, so $\deg(X)\nmid\deg(Y)$, showing the sharpness at $m=N/2$ in Theorem B (Li, 2012).

Example Variety	$n$	$m$	$\deg(X)$	$\deg(Y)$	Outcome
Complete intersection, 2 quadrics	$n$	$m = n/2$	--	--	Conclusion fails
$G(1,4) \subset \mathbb{P}^9$	$6$	$4$	$5$	$2$	Division fails at $m=N/2$

5. Quadratic Varieties and Swept-out Classifications

A subvariety $X \subset \mathbb{P}^N$ is quadratic if vanished identically under a collection of quadratic hypersurfaces. $X$ is swept out by $m$-dimensional quadrics through $x_0 \in X$ if for a general $x \in X$ there exists an $m$-dimensional quadric $Q_x \subset X$ containing $x_0$ and $x$.

Theorem C (Classification of Swept-out Quadrics):

Let $X \subset \mathbb{P}^N$ be a nondegenerate smooth quadratic variety of dimension $n$. If for some $x_0 \in X$, $X$ is swept out by quadrics of dimension $m \geq \lfloor n/2 \rfloor + 1$ passing through $x_0$, then $X$ is projectively equivalent to one of:

• (a) A quadric hypersurface in $\mathbb{P}^{n+1}$;
• (b) The Segre threefold $$\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$$;
• (c) The Plücker embedding $G(1,4) \subset \mathbb{P}^9$;
• (d) The 10-dimensional spinor variety $S_{10} \subset \mathbb{P}^{15}$;
• (e) A general hyperplane section of (b) or (c) (Li, 2012).

The proof employs Theorem A and classical characterizations (e.g., Hartshorne–Ionescu–Russo), and leverages prior results by Sato, Kachi–Sato, and Fu on varieties swept out by large-dimensional linear spaces or quadrics.

6. Significance in the Broader Geometric Landscape

The structure theorems governing subvarieties of smooth projective varieties with small codimension, as in Theorems A and B, constrain their geometry by enforcing divisibility in degree and conditions on the span. These results not only delineate possible subvarieties but also enable classification results, such as the precise description of quadratic varieties swept out by high-dimensional quadrics. The sharpness of all bounds underscores a tight interplay between the ambient projective geometry and the intrinsic geometry of the variety. The foundational results cited—Lefschetz, Barth–Larsen, and Hartshorne–Ionescu–Russo—highlight the cohomological and intersection-theoretic nature of these rigidity phenomena, reflecting a deep synthesis of topological, algebraic, and geometric methods (Li, 2012).