# Tag Info

### Quotients of schemes by connected groups

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$It seems that J. Kollár gave some examples in his paper "Non-quasi-projective moduli spaces". In pg. 1080, I think that there is an ...
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### Normalizer of solvable linear group is an algebraic group?

The normalizer of every connected closed subgroup of $\mathrm{GL}_n(\mathbf{R})$ is the stabilizer of its Lie algebra, so is Zariski-closed.
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### Question regarding the definition of linearization of line bundles

I think that you should regard the first definition as an imprecise version of the second definition. For example, suppose that $X$ is a point and so $L$ is simply a 1-dimensional vector space. ...
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### Frobenius and regular scheme

This is true Zariski-locally by Popescu's desingularisation theorem [Tag 07GC]. Indeed, any regular $\mathbf F_p$-algebra $A$ is geometrically regular [Tag 0381], so by Popescu's theorem it can be ...

### Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

I will answer even though this question was asked a year ago. Hopefully my answer is of some help. Yeah that's right. I don't quite understand the question, the Remark 2.7 says that any transverse ...

### Birational morphism that is not successive blow-down along smooth centers?

Let $C$ be a general curve of genus $g$ and take $\pi: Sym^g(C) \to Jac(C)$. Then brill-noether says that that fibers of dimension $r$ arise from $h^0=r+1$ line bundles which form a $g-h^0 h^1$ ...
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1 vote

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Accepted

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1 vote
Accepted

### What are the finite étale coverings of a quasi-hyperelliptic surface?

From pp. 488-489 of W.E. Lang's thesis (Ann. ENS. vol. 12, 1979) it follows that neither case can arise. For the fibration $X\to C$ has an elliptic base and all its geometric fibers are cuspidal ...
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Accepted

### Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

Here is a proof using a bit of commutative algebra and algebraic geometry, elaborating on comments by @abx and @ChrisWuthrich. The ring $$R=\mathbb{C}[x,y]/(x^3+y^3-1)$$ is a Dedekind domain, because ...
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### Is there an isotrivial elliptic surface of positive rank having a section of order $3$?

Let $k$ be a field and let $n\geq 3$ be an integer which is invertible on $k$. The following lemma will allow you to conclude that an elliptic curve over $k(t)$ which is isotrivial with $j$-invariant ...
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### A simple question about a statement of Kahler Manifold and Moishezon Manifold

The answer probably depends a lot on how exactly you define a Kähler manifold as there are multiple equivalent definitions. In any case, the definition of a positive line bundle is that it has a ...
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There is a general fact that a proper affine morphism is finite, and a general trick that for $\overline{X} \to S$ proper containing an open subset $X\to S$ affine, a closed subscheme $Z$ of \$\...