# 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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### Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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### Is there any product formula for local zeta function?

Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the ...
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### Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
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### A scheme is affine iff the natural map $X\to \operatorname{Spec}\Gamma(X)$ is an isomorphism

We know that the functor $\operatorname{Spec}: \mathsf{Rings}^{\text{op}}\to \mathsf{Schemes}$ is right adjoint to the global section functor $\Gamma: \mathsf{Schemes}\to \mathsf{Rings}^{\text{op}}$. ...
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### If $X(F) \cap Y$ is dense in $Y$, then $Y$ is defined over $F$.

Let $k$ be an algebraically closed field, $F$ a subfield of $k$, $A$ a finitely generated, reduced $k$-algebra, and $A_0$ an $F$-subalgebra of $A$, of finite type over $F$, such that the canonical $k$-...
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### Question regarding proof that a finite morphism is proper.

We know that a morphism $f: X \to Y$ is proper if and only if $Y$ can be covered by open subsets $V_i$ such that $f^{-1}(V_i) \to V_i$ is proper for each $i$. If a morphism is finite we can cover $Y$...
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### Is it always true that $N_{(G,G)}(T_1) \subseteq N_G(T)$?

Let $G$ be a connected, reductive linear algebraic group whose semisimple rank is $1$. Then $H := (G,G)$ is a connected semisimple group of rank one. Let $T_1$ be a maximal torus of $H$, and let $T$ ...
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### Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
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### Closed-form formula for system of two bivariate quadratic polynomials

Given a system of two bivariate quadratic polynomials: \begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 &= 0 \\ b_0 + b_1 x + b_2 y + b_3 xy+b_4 x^2 + b_5 y^2 &= 0 \end{...
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### Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
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### how to calculate the implicit Cartesian equation from calabi yau threefold?

i find mathematica version about parametric form, how to calculate the implicit Cartesian equation from calabi yau threefold? ...
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### What is the definition of the dimension of an algebraic manifold?

I have a very basic question. It says on Wikipedia that an algebraic manifold is an algebraic variety which is also a manifold. So suppose I have an algebraic manifold $V$ which is an affine variety ...
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### Base point free linear system

Let $X$ be a (compact) Riemann surface. Let $D$ be a divisor. In Rick Miranda's book on Riemann surfaces, on page 160, there is a bijection between Base-point-free linear systems of dimension $n$ on ...