判断集合X为仿射、拟仿射还是两者皆非？X=Q-L分析求助

Great job getting to the point where you've identified $X = Q - L \cong \mathbb{A}^1 \times \mathbb{P}^1$—that's a key milestone! Now we can leverage the well-understood properties of this product variety to thoroughly characterize $X$'s type and key features:

1. Core Classification: Smooth Quasi-Projective Rational Surface #

• Smoothness: The quadric $Q \cong \mathbb{P}^1 \times \mathbb{P}^1$ is a smooth projective surface, and removing the smooth line $L$ (a closed subvariety of $Q$) preserves smoothness. So $X$ is a smooth 2-dimensional quasi-projective variety.
• Rationality: Both $\mathbb{A}^1$ and $\mathbb{P}^1$ are rational (each is birational to $\mathbb{P}^1$), and products of rational varieties are rational. This means $X$ is a rational surface—it’s birational to $\mathbb{P}^2$, the simplest projective surface.

2. Picard Group (Line Bundle Classification) #

The Picard group of a product relates directly to the Picard groups of its factors:

• $\text{Pic}(\mathbb{A}^1)$ is trivial (the affine line has no non-trivial algebraic line bundles).
• $\text{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$, generated by the hyperplane class $\mathcal{O}(1)$.

Thus, $\text{Pic}(X) \cong \text{Pic}(\mathbb{A}^1 \times \mathbb{P}^1) \cong \text{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$. The generator is the pullback of $\mathcal{O}(1)$ from the $\mathbb{P}^1$ factor.

3. Kodaira Dimension (Complexity of Canonical Bundle) #

Kodaira dimension measures how "rich" the sections of the canonical bundle are:

• $\kappa(\mathbb{A}^1) = -\infty$ (affine varieties have trivial canonical bundles that don’t produce non-trivial sections in any positive degree).
• $\kappa(\mathbb{P}^1) = -\infty$ (the canonical bundle of $\mathbb{P}^1$ is $\mathcal{O}(-2)$, an anti-ample bundle with no non-trivial sections).

For products where both factors have Kodaira dimension $-\infty$, the product also has $\kappa(X) = -\infty$—this aligns with $X$ being a rational surface (all rational surfaces have Kodaira dimension $-\infty$).

4. Ruled Surface/Affine Bundle Structure #

$\mathbb{A}^1 \times \mathbb{P}^1$ is the trivial affine bundle over $\mathbb{P}^1$: if you project onto the $\mathbb{P}^1$ factor, every fiber is isomorphic to $\mathbb{A}^1$. It’s also an open dense subset of the trivial ruled surface $\mathbb{P}^1 \times \mathbb{P}^1$ (your original quadric $Q$), since $X = (\mathbb{P}^1 \setminus {pt}) \times \mathbb{P}^1$.

5. Affine vs Projective Status #

Important note: $X$ is not affine. Even though $\mathbb{A}^1$ is affine, the product with $\mathbb{P}^1$ (a projective variety) can’t be affine. Instead, it’s a quasi-projective variety—an open dense subset of the projective quadric $Q$.

To wrap up, $X$ is a smooth, quasi-projective, rational surface with Picard group $\mathbb{Z}$, Kodaira dimension $-\infty$, and structure as a trivial affine bundle over $\mathbb{P}^1$.

内容的提问来源于stack exchange，提问作者Tensor_Product