# Tagged Questions

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. ...

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### Algebraic surface with infinitely many exceptional curves

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point. If I understood correctly every surface $X$ ...
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### Automorphisms of $\mathbb{C}[x_1, \dots, x_n]$

Are the linear transformations, and the automorphisms of the form $\sigma(x_1, \dots, x_n) = (x_1 -f(x_2, \dots, x_n), x_2, \dots, x_n)$, where $f$ is a polynomial, generators of the group of ...
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### Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
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### Identifying two points on an algebraic curve

Given a smooth algebraic curve $C$, say projective over an algebraically closed field $k$, is it always possible to identify two distinct closed points $x, y$ on $C$ to produce a curve with a single ...
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### Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
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### Maximal ideals of polynomial ring

We know that if $k$ is algebraically closed, then each maximal ideals of $k[x_1, x_2, \ldots , x_n]$ are of the form $(x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n),$ where $a_1, a_2, \ldots , a_n \in k$ (...
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### Hartshorne generically finite morphisms (II, 3.7)

I have a question concerning one of the exercises of Hartshorne, Ch. II. Namely: Exercise 3.7 about gerneically finite morphisms. A morphism $f: X \rightarrow Y$ with Y irreducible and $\eta$ ...
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### Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
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### definition of singular locus of a variety

Given a variety $X$ over $k$, we can consider which points are regular, and we can define the singular locus $\operatorname{Sing}(X)$ as the complement of the regular points in $X$. My question is, ...
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### affine scheme that is finite type over $\mathbb{Z}$

I have an affine scheme that is finite type over $\mathbb{Z}$, so by definition I can cover this Spec $A$ by Spec $B_i \ (1 \leq i \leq n)$ where each $B_i$ is a finitely generated $\mathbb{Z}$ ...
### True or False: $f$ is injective if and only if $f^*$ is surjective where $f^*$ is corresponding to the pullback to $f$.
Let $f: X\rightarrow Y$ be a morphism of affine varieties and $f^*: A(Y)\rightarrow A(X)$ the corresponding homomorphism of the coordinate rings. The question is whether this is true or false: $f$ is ...