# Questions tagged [projective-schemes]

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

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### projective space minus a closed point

Let $k$ be an algebraically closed field and let $\mathbb P^n_k=\text{Proj}(k[x_0,x_1,...,x_n])$ . If $n\ge 2$, and $p\in \mathbb P^n_k$ is a closed point, then can $\mathbb P^n_k\setminus \{p\}$ be a ...
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### The function field of the projective scheme $\mathbb{P}^{r}_{k}$.

Let $r\in\mathbb{Z}_{> 0}$, let $k$ be a field and $X=\mathbb{P}^{r}_{k}$ and $S=k[X_{0},...,X_{r}]$. By definition we have that the function field is defined as $K(X)=\mathcal{O}_{X,\eta}$ where ...
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### Plane minus a projective line vs. plane minus a conic

Let $k$ be a field. In $\mathbb{P} = \mathbb{P}^2(k)$, an irreducible conic $C$ is isomorphic to a projective line $U = \mathbb{P}^1(k)$ (which we take to be a line in $\mathbb{P}$). If I am not ...
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### Separating Points and Tangent Vectors (real curves)

In [Hartshorne, Proposition 7.3.] as well as in [Görtz & Wedhorn, Rem. 13.55] and [Vakil Notes, around 19.2] the following is said: If $X$ is a curve over (let's say) $\mathbb{C}$ (algebraically ...
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### Calculate $\dim_k(M_d)$ and $\dim_k \Gamma(X; \tilde{M} \otimes \mathcal{O}_X(d))$ for every $d \in \mathbb{Z}_{\ge 0}$.

I' trying to solve the following problem: Let $k$ be a field, and let $Z$ be the $k$-scheme $Spec(k \times k) = Spec(k) \sqcup Spec(k)$. Let X = $\mathbb{P}^1_k$. The free rank-one $k \times k$-...
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### Is the preimage of the generic point of the map $\mathbb{P}^n_R \to \operatorname{Spec}R$ open?

Let $R$ be an integral domain and let $z$ be the generic point of $\operatorname{Spec}R$. Let $\pi: \mathbb{P}^n_R \to \operatorname{Spec}R$. I want to show that $\pi^{-1}(z)$ is open. Any comments ...
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### Normal linear system (embeddings into projective space)

We consider a variety $X$ over $k$. I have a question on an statement from wikipedia deals with interpretations of normality in algebraic geometry. It says: An older notion is that a subvariety $X$ ...
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### If $R = k[T_0,\cdots,T_n]/I$ is integral, then $\text{dim Proj R}=\text{dim D}_+(T_i)$ for some $i$

Why is it that if $R = k[T_0,\cdots,T_n]/I$ is integral, $k$ a field, then $\text{dim Proj R}=\text{dim D}_+(T_i)$ for some $i$? $D_+(T_i)$ is just the set of primes of $\text{Proj R}$ which don't ...
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