# Questions tagged [projective-schemes]

This tag is for questions relating to "projective scheme".

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### Morphisms from laurent series to Projective space

Let $k$ be a field and $k((x))$ be its field of formal laurent series. I'm trying to understand morphisms $f : Spec \ k((x)) \rightarrow \mathbb{P} = Proj \ k[T_0, T_1]$. For example, does specifying ...
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### Maps of graded rings inducing isomorphism between Proj.

Let $\Bbbk$ be a field. Let $S,T$ be graded $\Bbbk$-algebras, and let $\phi:S\to T$ be a graded ring map. We can assume finite generation if things become simpler. We have that if for sufficently ...
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### Is $\mathbb{P}^1$ a group scheme?

It's known that the set of meromorphic functions (functions to $\mathbb{C}\cup\{\infty\}$) on a complex variety $X$ forms a field, called the function field of $X$. Edit: Thanks to the comment by @...
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### For divisor $D$, how to describe isomorphisms $H^0(X,\mathcal{O}_D(nD)) \xrightarrow{\sim} H^0(X,\mathcal{O}_D\left((n+1)D\right)$?

Let $X$ be a scheme, $D$ an effective divisor on $X$ with structure sheaf $\mathcal{O}_D$, and $U = X\setminus D$. I think I need $D$ either ample or affine. If necessary we can assume $\mathcal{O}_X$ ...
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### Algebraic Projective Noether normalization

Reading through the literature I see that there are two proofs for the Noether normalization for projective varieties (over infinite fields). The first, "geometric", by iterating a ...
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### Global sections of a torsion sheaf

Let $X$ be a smooth projective curve, $\mathcal{F}$ a torsion sheaf on $X$, $\Gamma(X,\mathcal{F})$ finitely generated by $\{s_1,\ldots,s_n\}$ (or $n:= \dim \Gamma(X, \mathcal{F})$). Is $\mathcal{F}$...
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### Sections of a torsion sheaf

I am trying to get a better understanding of torsion sheaves over projective schemes. Is it true that if $\mathcal{F}$ is torsion over a projective scheme $X$, then \Gamma(X,\mathcal{F}) \cong \...
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In following all schemes $X$ will be considered as separated, of finite type over an algebraically closed field $K$ of characteristic $0$. Recall that a Hilbert function $HF_S: \mathbb{N}_0 \to \... 0 votes 1 answer 75 views ### Definition of$ \ \mathcal{O}_{\mathbb{P}_k^n}(1)$[duplicate] I've found references to the object$\ \mathcal{O}_{\mathbb{P}_k^n}(1), \ $where$\mathbb{P}_k^n \ $is the usual scheme, but I'm not able to find its proper definition. Is it just a line bundle ... 3 votes 1 answer 100 views ### Show that the dimention of the intersection of projective linear sub-spaces of dimentions$d_1$and$d_2$of$\mathbb{P}^n$is bigger than$d_1+d_2-n$Proposition: Let$L$and$M$linear projective subspaces of$\mathbb{P}^n$of dimention$d_1$and$d_2$respectively. Prove that$\operatorname{dim}(L\cap M)\geq d_1+d_2-n$(we consider the dimention ... 1 vote 1 answer 318 views ### Does the degree of a projective curve depend on the embedding into projective space? I'm studying the classification of curves in$P^3$in chapter IV of Hartshorne. I've already studied chapter I, chapter II (sections 1 to 6), and chapter III (sections 1 to 7) of Hartshorne in my ... 0 votes 0 answers 238 views ### nth Veronese subring embedding induces isomorphism of schemes (Vakil FOAG 6.4.D) Exact exercise is: show the embedding map of graded rings$S_{n\bullet} \to S_\bullet$for$S_\bullet$finitely generated graded ring induces an isomorphism$\operatorname{Proj} S_\bullet \to \...
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Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a ...