# Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

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### If $D$ is Cartier and $D_{\overline{k}}$ is $Gal(\overline{k}/k)$-invariant, is it also Cartier?

Suppose we have a proper scheme (or even a variety) $X$ over a non-closed field $k$. Take a ($\mathbb{Q}$-)Cartier divisor on $X$ such that $D_{\overline{k}}$ is $Gal(\overline{k}/k)$-invariant. Is it ...
1 vote
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### Prove the intersection number of a irreducible curve and its hessian at a simple node is 6

I am trying to solve problem 7.22 in Fulton's Algebraic Curve. In the first part of the question, we are asked to prove the intersection of an irreducible plane curve $F$ and its hessian $H$ is 6 at a ...
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### Stabilizer of a subspace of a representation of an affine group

In Chapter VIII, Prop 12.1, of this note by JS Milne (as well as Chapter 4, Prop 4.3, of his book, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field), the following definition ...
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### Hartshorne exercise I.2.7 reduction

How are you supposed to use Hartshorne Ex I.2.7(b) in Ex I.2.6? The problem is to prove dimension of a quasiprojective variety is the same as dimension of its closure in $\mathbb{P}^n$. The previous ...
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### How to define $G$-equivariant module over a $G$-equivariant algebra

Let $k$ be a field and $G$ be a finite group. Furthermore, we assume the action of $G$ on $k$ is trivial. In this case, a $G$-equivariant sheaf $V$ on $k$ is just a $k$-vector space $V$ with $G$-...
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### Tensor algebra and Universal enveloping algebra

Let $\mathfrak g$ be a Lie algebra which is NOT reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
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### Closure of orbit of Borel subgroup $B_n$ acting on triangular matrices $T_n$.

Let $\Bbbk$ be an infinite field. Consider the set of $n\times n$ upper-triangular matrices $T_n$. Let $B_n$ be the set of invertible $n\times n$ upper-triangular matrices. Consider the following ...
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1 vote
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### Bidegree genus formula for curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I've been going through some online notes trying to learn a first course in algebraic geometry. Currently we are looking at the theory of curves. By convention of the notes, these are assumed to be ...
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### Connectedness implies equivariance

In theorem 6.7.2 in the note Moduli and Stacks by J.Alper, we are given the following: Let $X,U$ be algebraic spaces of finite type over an algebraically closed field $k$. Let $G$ be a connected ...
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### Dimension of fibre product

Let $f:X \rightarrow Y$ is dominant morphism of schemes with $Y$ irreducible. Let $p:Z \rightarrow Y$ be quasi finite morphism of schemes such that $\overline{p(Z)} \ne Y$. Let $Z’$ be fibre product ...
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### Which $f \in \mathbb{C}[x,y]$ satisfies $\mathbb{C}[x,f] = k(x,f) \cap k[x,y]$? [closed]

Let $f=f(x,y) \in \mathbb{C}[x,y]$ with $\deg_x(f) \geq 1, \deg_y(f) \geq 1$. Question: Which $f \in \mathbb{C}[x,y]$ satisfies $\mathbb{C}[x,f] = \mathbb{C}(x,f) \cap \mathbb{C}[x,y]$? Of course, ...
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### Proving that $R^if_*\mathbb{C} \otimes_{\mathbb{C}} \mathcal{O}_S \simeq R^if_*(f^{-1}\mathcal{O}_S)$

I'm reading "Introduction to Hodge Theory", by Bertin, Demailly, Illusie and Peters and I'm stuck trying to understad this identification. Let $f: X \rightarrow S$ be a smooth and proper map ...
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1 vote
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### Additivity of Euler characteristic on long exact sequence [closed]

In the book of Vakil, the Foundation Of Algebraic Geometry, there is an exercise (19.4.A) that I can't do and I'd like to have some help. The exercise is the following : We take a projective $k$-...
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