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Let me reproduce the relevant bit from the reference above (Internet Archive): Exercise 3.1. Let $X$ be a variety of general type. Prove that a subvariety of $X$ passing through a general poin …

Edit: I am working over $\mathbb C$ here, but a similar answer work over an arbitrary algebraically closed field. See my comment below as well as Andrea Ferreti's. The degree of the divisor is equal …

The terminology is fairly common in classical works on projective algebraic/differential geometry. I am not aware of its origins. Anyway it is used rather informally and only means that you have a $k$ …

If $X\subset \mathbb P^n$ is a subvariety of dimension $m$ embedded by a linear system $V \subset H^0(X,\mathcal O_X(D))$ then the degree of $X$ is equal to $D^m$.

I will work over $\mathbb C$. Although I have not checked, the example below should work for characteristic different from $3$. To exhibit a degree $d$ projective surface $S \subset \mathbb P^3$ not …

The existence of a morphism of a compact Kähler manifold to curve of genus $g \ge 2$ is a purely topological fact. This was first proved by Siu, and rediscovered independently by Beauville. The preci …

Let me give some remarks assuming that $k=\mathbb C$. a. The intersection $T_p V \cap V$ must have codimension one. Otherwise $H^0(\mathbb C^n,\mathcal O_{\mathbb C^n})$ would inject into $H^0 (V,\ma …

Indeed much more is true. Suppose $X$ is an irreducible algebraic manifold admitting a transitive action of a linear algebraic group $G$. If $Y$ and $Z$ are irreducible subvarieties of $X$ then for …

Hyperquadrics of dimension at least three.

It seems impossible to give a better answer than Pete L. Clark's but those passing by might appreciate a sketchy proof. If $X \subset \mathbb P^n$ is a projective variety then it is the intersection …

Perhaps it is better to phrase the question in terms of Lie algebras. For instance, if you want to know which are the possible codimension one Lie subalgebras of a given finite dimensional Lie algebra …

Perhaps this paper of Harris is what you are looking for: A bound on the geometric genus of projective varieties. It generalizes Castelnuovo's bound to smooth projective varieties of arbitrary dimens …

Everything below is defined over $\mathbb C$. Let $T$ be a smooth affine variety, $\pi : \mathscr X \to T$ be a smooth family of smooth projective varieties, and $\mathcal F$ be a locally free cohe …

In the analytic category there are line-bundles over $X = \mathbb C^2 - \{ 0\}$ which do not extend to $\mathbb C^2$. Since $X$ has the homotopy type of the sphere $S^3$, the exponential sequence $$ …

Let $X$ be a smooth complex projective variety. Let $Y$ be a smooth divisor on $X$, and let $\mathfrak X$ be the formal completion of $X$ along $Y$. Question. If $\mathcal L$ is a line bundle on …