# 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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### The cone over a projective variety

I'm trying to prove that $I(C(Y))=I(Y)$, where $C(Y)=\pi^{-1}(Y)\cup \{(0,\ldots,0)\}$ the cone over $Y$ and $\pi:\mathbb A^{n+1}-\{0,\ldots,0\}\to \mathbb P^n$ the projection which sends the point ...
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### Making sense out of $F$-structures and the notion of $F$-variety

For almost two years I have been trying to make sense out of several claims about varieties over nonalgebraically closed fields made in the first chapter of the textbook Linear Algebraic Groups by T.A....
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### About the definition of polynomials on vector spaces?

In the book Linear Systems Theory and Introductory Algebraic Geometry (R. Hermann) the author defines A polinomial on V (a $\mathbb K$-vector space) is an element of the smallest ...
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### A question of terminology regarding exceptional curve or is it divisor.

So I kept on reading the book by Griffiths and Harris called Principles of Algebraic Geometry and I've seen a definition of exceptional divisor of the first kind. On page 487: A smooth rational ...
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### must this extension of a DVR be unramified?

Let $A$ be a normal domain, and $P$ a height 1 prime, then $A_P$ is a DVR. Let $K$ be the fraction field of $A_P$, and let $L$ be a finite Galois extension of $K$ of degree $e$, let $B$ be the ...
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### Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point

I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'. My question concerns Definition 1.27: Let $A$ be a Noetherian ...
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### Showing $\exp:\mathscr{O}_X\to\mathscr{O}_X^*$ is an epimorphism of sheaves

$\newcommand{\O}{\mathscr{O}}$Let $X=\Bbb{C}$. Define $\O_X$ to be the sheaf of holomorphic functions, and $\O_X^*$ to be the sheaf of invertible (i.e. nonvanishing) holomorphic functions, the latter ...
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### Quotient field of ring of regular functions at some point in affine variety is the field of rational functions on the variety

I am reading Hartshorne's book of Algebraic Geometry. I am stuck in understanding why quotient field of the local domain O_p (where O_p denotes the ring of regular functions at a point p in affine ...