Questions tagged [projective-schemes]

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

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48 views

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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51 views

Projective space over the field of rational functions determined by generic point

In the situation of Ex. II. 6.1 in Hartshorne, let $\eta_X$ be the generic point of $X$ where $X$ is a locally factorial noetherian separated integral scheme. I am trying to understand why $\mathbb P_{...
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53 views

An exact sequence for coherent sheaves on $\mathbf{P}^n_k$

Let $k$ be a field and $\mathscr{F}$ a coherent sheaf on $\mathbf{P}^n_k$. In paragraph $5.2$ of Fundamental Algebraic Geometry, it is claimed that if $H\subseteq\mathbf{P}^n_k$ is a hyperplane which ...
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47 views

When is $\operatorname{Proj}k[x,y,z]/(x^n+y^n+z^n)$ is a regular scheme?

When is $\operatorname{Proj}k[x,y,z]/(x^n+y^n+z^n)$, with $n\geq 1$ and $k$ an algebraically closed field, a regular scheme? From Liu p135, the answer is '$n$ is prime to $ch(k)$'. I tired to use ...
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74 views

Hilbert polynomial of the blow-up of a projection

Let $p:\mathbb{P}^r\dashrightarrow\mathbb{P}^{r-k-1}$ with $0\le k<r$ be the projection $(a_0:\cdots:a_r)\longmapsto (a_{k+1}:\cdots:a_r)$. Such a rational map has base $K:=V(X_{k+1},\cdots,X_r)\...
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1answer
40 views

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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29 views

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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55 views

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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53 views

For any ideal $I$, $V_+(I)=V_+(J)$ for a specific homogeneous $J$

Let $S=\bigoplus_{n\geq 0}S_n$ be a graded ring and $S_+:=\bigoplus_{n\geq 1}S_n$. We define $\text{Proj}(S)$ as the set of homogeneous prime ideals of $S$ not containing $S_+$ and, for any ideal $I\...
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78 views

Twisted sheaves in Hartshorne

I have a few questions concerning twisted sheaves as defined in Hartshorne, II.15. 1) Let $X = \text{Proj}(S)$ for a graded ring $S$. I do understand how Hartshorne defines the sheaves $\mathcal O_X(...
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76 views

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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29 views

Cech - cocycle on the projective line over $\mathbb{F}_2$

I want to solve the following exercise. Consider the projective line $X = \mathbb{P}^1_{\mathbb{F}_2} = \rm{Proj}_{\mathbb{F}_2}\mathbb{F}_2[t_0,t_1]$ over the field $\mathbb{F}_2=\mathbb{Z}/2\...
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1answer
65 views

$X$ a $k$-variety and given a DVR $R$, $Spec(R) \rightarrow X$ maps the closed point of $R$ to a closed point of $X$

Let $X$ be a smooth projective $k$-variety. I want to prove that if we have a $k$-morphism : $$ f : \operatorname{Spec}(R) \rightarrow X$$ where $R$ is a DVR, which is compatible with the following ...
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170 views

Question about Theorem 1 Chapter II.8 from Mumford's red book

Let $R$ be a valuation ring with maximal ideal $M$ and $L$ its residue field. We let $k$ be the fractions field which we assume to be algebraically closed. Let $\mathcal{L}$ be a closed subset of $\...
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46 views

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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59 views

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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2answers
89 views

Projective $n$ - space is not affine over any ring $R$

A similar question has been asked here already, but there was no final answer to the problem in the most general case. I wish to show that: For $n>1$ and a ring $R$, the projective $n$ - space $\...
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1answer
50 views

Nonreduced quadric hypersurfaces

I am reading a text in which one speaks of quadric hypersurfaces $\mathcal{Q}$ (in $\mathbb{P}^n(k)$ with $k$ a finite field) which are "everywhere nonreduced." What does this mean ? (And is there a ...
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1answer
33 views

Geometry of taking the sheaf of algebras associated to a line bundle.

Given a line bundle $L$ on a scheme $X$, we can construct the sheaf of $O_X$ algebras $\bigoplus\limits_{n=0}^\infty L^n$, which by the global Spec functor induces a map from some other scheme to $X$. ...
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44 views

EDITED: What is the correspondence between the associated sheaves and subschemes of Weil prime divisors?

I'm interested in the correspondance between the Picard group and the Weil group. Over a sufficiently well-behaved scheme $X$, like a projective curve or if $X$ is regular, these are isomorphic. The ...
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32 views

Comparison between degree of $kX$ and $X$ for a projective variety $X$ and the power map $k$.

Let $k:\mathbb{P}^{n}\to \mathbb{P}^{n}$ be the map defined by $(x_0:\dots:x_n)\mapsto(x^k_0:\dots:x^k_n)$, and let $X\subset \mathbb{P}^{n}$ be a projective variety. Then kX (Not the pushforward ...
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1answer
42 views

If $S=A[T_0,\cdots,T_d]$, $X=\operatorname{Proj}S$, then $O_X(n)(X)=S_n$ if $n\geq0$, $0$ otherwise.

Here, $A$ is just some ring, $d>0$ In the following, $O_X(n)=\widetilde{S(n)}$, and $S(n)$ is $S$ with the graduation shifted (i.e. $S(n)_d=S_{n+d})$ and $S_{(f)}$ are elements of degree $0$ of $...
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1answer
29 views

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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23 views

Is pullback of ample sheaf by formal completion ample again?

Suppose $X$ is a Noetherian scheme and $I$ a quasi-coherent sheaf if ideals. We can formally complete the original scheme to get a formal scheme and a morphism of locally ringed spaces $k:\mathcal{X} \...
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44 views

Čech cohomology of a subscheme defined by a homogeneous polynomial

This is Hartshorne III.4.7 Let $k$ be a field and let X be a subscheme of $\mathbb{P}^2_k$ defined by a single homogeneous polynomial $f \in k[x,y,z]$ of degree $d$. Assume that $(1: 0: 0) \not \in X$...
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1answer
107 views

Constant morphism from $\mathbb{P}^m_k$ to $\mathbb{P}^n_k$ as schemes [duplicate]

Let $k$ be a field and $m>n \in \mathbb{N}$. Then any morphism of schemes $\mathbb{P}^m_k \rightarrow \mathbb{P}^n_k$ is constant. What I know is that every morphism from a scheme $X$ to $\mathbb{...
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1answer
59 views

Čech cohomology of $\mathbb{P}^2_k \setminus [1:0:0]$

We can cover $X=\mathbb{P}^2_k \setminus [1:0:0]=D_+(y) \cup D_+(z)$ and the Čech complex is given by $$ 0 \rightarrow \mathcal{O}_X(D_+(y)) \oplus \mathcal{O}_X(D_+(z)) \rightarrow \mathcal{O}_X(D_+(...
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1answer
79 views

Proof Explanation of Lemma III 7.4 Hartshorne

In Lemma III 7.4 of Hartshorne we have $X$ is a closed subscheme of co-dim $r$ in $P=\mathbb P_k^N$. $\mathscr F$ is a coherent sheaf on $X$. Then $\mathscr F$ is naturally an $\mathcal O_P$ module. ...
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34 views

Notion of depth of coherent sheaf on projective variety $X \subseteq \mathbb P^n_k$

If $R$ is a $\mathbb N_{\ge 0} $ - graded commutative Noetherian ring with $R_0=k$ a field and $\mathfrak m=\oplus_{i>0} R_i$ , and $M$ is a finitely generated $\mathbb Z$-graded $R$ module , then ...
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1answer
85 views

Constructing projective Calabi-Yau varieties

A nice way of constructing a non-projective Calabi-Yau threefold is to take the total space $$Y:= \mathrm{Tot}(\omega_S) = \mathbf{Spec}(\mathrm{Sym}^\bullet(\omega_S^\vee))$$ of the canonical ...
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90 views

Interpretation of Complex disc around origin as a variety

Can we think of any complex disc around origin(say $\Delta$) as a projective variety? I am asking this question because I have encountered the product $\mathbb{P}^{n} \times \Delta$ in an algebro-...
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71 views

Global sections of a line bundle

let $X$ be a normal projective curve over field $k$. let $Z=\sum_{x \text{ closed}} n_x [x]$ be a $0$-cycle on $X$ and $D \in Div(X)$ a Cartier divisor such that $[D]=Z$. then $D$ gives rise for a ...
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1answer
103 views

How to prove $\mathcal O_X(m)\otimes_{\mathcal O_X} \mathcal O_X(n)= \mathcal O_X(m+n)$?

Let $B$ be a graded ring and $X=\mathrm {Proj}\: B$. Let $B(m)$ denote the twist of $B$ (i.e. a graded $B$-module such that $B(m)_d=B_{m+d}$,) $\mathcal O_X(m)$ denote the quasi-coherent sheaf $\tilde{...
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1answer
238 views

Quasi-coherent sheaf on $Proj\ S$

Given a graded ring $S$ and a quasi-coherent sheaf $\mathcal{F}$ on $Proj\ S$, does there exist a graded $S$-module $M$ such that $\mathcal{F}\cong \widetilde{M} $? I know the result is true when $S$...
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1answer
119 views

Understanding $\mathbf{Proj}$

I have a question concerning the construction of $\mathbf{Proj}$. We take a graded quasicoherent sheaf $\mathscr{R} = \bigoplus \limits_{n=0}^\infty \mathscr{R}_n$. Next for all affine subsets $U = \...
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31 views

Degree of projective curve related to its reduction

Let $X$ be an irreducible closed subscheme of dimension $1$ of $\mathbb{P}^n_k$ for some $n$. Let $X'$ be the closed reduced subscheme associated to it, which is an integral projective curve. ...
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93 views

Grothendieck type vanishing result for Local Cohomology over not necessarily affine schemes?

Let $(X,\mathcal O_X)$ be a Noetherian, affine Scheme and $\mathcal F$ be a quasi-coherent Sheaf of $\mathcal O_X$-modules on $X$. Let $\dim \mathcal F$ be the Krull dimension of $\{x\in X| \mathcal ...
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147 views

Definition of scheme-theoretic multiplicity

In Gathmann's notes on algebraic geometry (https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf), he remarks in Lemma 6.1.4. (page 93) that the length of a zero-dimensional ...
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34 views

Relation between Zero dimensional subschemes and saturated ideals

Why is studying zero dimensional subschemes (for simplicity let's say closed subschemes) of $\mathbb{P}^{n}_K$ equivalent to studying saturated homogeneous ideals $I$of $K[x_0,...x_n]$ with krull ...
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173 views

Morphism between projective spaces

I have the following problems of morphisms between projective spaces, I would like to know if someone has any hint or knows how to solve them: a) Suppose that there exists $ f: \mathbb{P}^r \...
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1answer
53 views

Graded prime ideals of infinite poylnomial ring

Consider the infinite polynomial ring, $A$ unital commutative, $$S= A[x_1,x_2,\ldots ].$$ We give the rings the grading with $\deg x_i=1$. $\operatorname{Proj}S$ denotes the homogeneous prime ideals ...
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1answer
118 views

Showing Proj $R[x,y]/(x^2)$ is not affine scheme

So I started reading the Proj construction. I wanted to get more understanding by consider the graded ring $$\frac{R[x,y]}{(x^2)}$$ where $x,y$ have degree $1$. Let $X= Proj(R[x,y]/(x^2))$. So I ...
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93 views

Projective space $Proj$

We let $S$ be a graded algebra. I have a 2 questions regarding the Proj construction. It seems to me that we do not know what $O_{Proj S}Proj(S))$ is ? How is the map $S_0 \rightarrow \Gamma(Proj S,...
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47 views

Projective subscheme as union of its components

This is the same question Writing a projective scheme as a union of irreducible subschemes. but this was not clearly answered: if I have $Z \subseteq \mathbb{P}^n_k$ a projective scheme, I can ...
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1answer
207 views

Degree of zero dimensional schemes in projective space

Background: Let $k$ be a field (I'm mostly interested in $k$ an algebraically closed field of characteristic $0$) and let $X$ be a zero dimensional closed subscheme of $\mathbb{P}^n_k$. Question: ...
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60 views

What is the multidegree of a curve $C \subset \mathbb{P}^n \times \mathbb{P}^m$?

What is the multidegree of a curve $C \hookrightarrow \mathbb{P}^n \times \mathbb{P}^m$? I'm reading Notes on stable maps and quantum cohomology by W. Fulton and R. Pandharipande, and on page 14, ...
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1answer
152 views

The map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$

Show that the map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ given by $(w, x, y, z) \mapsto (s^3, s^2t, st^2, t^3)$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$, ...
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1answer
241 views

Embed a weighted projective space into an unweighted projective space.

To show is the following. Let $X = P(a_0,\dotsc,a_n)$, $a_i \geq 1$ be a weighted projective space (that is $X = \operatorname{Proj} k[x_0,\dotsc,x_n]$, where $\operatorname{deg} x_i = a_i$). ...
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1answer
36 views

Statement in stack project on Relative Gluing

This is a screenshot of a statement from stacksproject. I have a number of confusion of this statement: What is a scheme $f_U:X_U \rightarrow U$ over $U$? Does this means that $X_U$ is ...
3
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1answer
205 views

Multiplicity and degree of irreducible projective subschemes.

Suppose $X \subset \mathbb{P}^n$ is an irreducible projective scheme. Then its multiplicity $\mu_X$ is defined as the length of the local ring $\mathcal{O}_{X,\eta}$ over itself, where $\eta$ is the ...

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