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Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension incr …

$\DeclareMathOperator{\Spec}{Spec}$ Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are redu …

The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways: The maximal dimension of the image of the rational maps $φ_{|mL|} : X \dashrighta …

Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$. Assuming $X$ is nondegenerate and irred …

Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $ \DeclareMathOperator{\Aut}{Aut} \Aut(X)$. Define a torus in $\Aut(X)$ to be a faithful alge …

The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as \begin{equation} \det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{ …