# 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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### How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed ...
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### Complementary Text to Gunning and Rossi - Analytic functions in several complex variables

I'm currently a second year student who has a background in group theory, ring theory, galois theory, metric spaces and point set topology. I'm currently taking courses in algebraic topology, advanced ...
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### Understanding $O_{\mathbb{P}(F)}(d)$ on $\mathbb{P}(F)$ (in local coordinates)

Say $F$ is a vector bundle of rank $r+1$ on some scheme $X$ with transition maps (cocycle) $\psi_{ij}$ (with respect to some open cover $U_i, U_j,\ldots, U_k$ of $X$). We denote by $\mathbb{P}(F)$ the ...
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Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. I heard that people say "an additive basis" of $\mathbb{C}[Gr(k,n)]$. What does additive mean? Thank you ...
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### Geometric equivalent of the degree zero divisor class group of an algebraic function field (in the singular case)

In an algebraic function field $F/k$ we have the degree zero divisor class group $\text{Cl}^0(F/k)$. Now since any such function field corresponds to the function field of a normal projective curve ...
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### What happens with negative plurigenus?

It is a well known result that for a smooth, projective k-variety, the dimension of the global section $H^0(X,K_X^j)$ of $j$-powers of the canonical bundle. Also called plurigenus, are birational ...
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### Handelman's Theorem on Nonnegative polynomial in Compact Polytope?

Background: Representing polynomials by positive linear functions on compact convex polyhedra by David Handelman Consider a polynomial $f\in K[x_1,\ldots,x_n]$ in a compact polytope, related ...
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### Noether Normalisation lemma

How do we explain Noether normalisation in terms of line bundles and linear systems? If $X$ is a smooth irreducible projective variety over an algebraically closed field of char 0 of dimension $r$. ...
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### What's the difference between $\mathbb{A}^n$ and $\mathbb{A}^{n+1}$?

Besides the obvious difference in topological dimension. If you want to distinguish between $\mathbb{R}$ and $\mathbb{R}^2$, take an open set in the plane, remove a point, then it's still connected. ...
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### Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
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### Polynomial in Compact Polytope: Algebraic Description for the Compact Polytope?

Consider a polynomial $f\in K[x_1,\ldots, x_n]$ where $K=\mathbb R$. For example $$f[x_1,x_2,x_3]=x_1 x_2+x_3$$ \begin{eqnarray*} x_{1} & \in & [0.2,0.5]\\ x_{2} & \in & [0,1]\\ x_{3}...
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### Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
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### Can I choose $k+1$ hypersurfaces to avoid a fiber of dimension $k$ in projective space?
Let $X$ be a closed subscheme of dimension $k$ in $\mathbb{P}^n_A$, where $A$ is a Noetherian ring. In Exercise 11.3.C of Ravi Vakil's notes, it is shown that one may choose $k+1$ hypersurfaces such ...
Let $\phi_1, \phi_2 : S \longrightarrow X$ be two sections of a finite étale cover $X \longrightarrow S$ of a connected scheme $S$. Assume that $\phi_1 \circ \overline{s} = \phi_2 \circ \overline{s}$ ...