Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [projective-schemes]

The tag has no usage guidance, but it has a tag wiki.

0
votes
0answers
40 views

Universal mapping property for projective schemes and globalising it to projective bundles

Let $Y = \text{Spec}A$ be a noetherian affine scheme. Let $S$ be a graded $A$-algebra which is finitely generated by $S_{1}$ as an $S_{0}$ algebra. In other words, $S$ looks like $$ S = A[x_{0}, x_{1},...
0
votes
0answers
38 views

On double points of a Projective space

$\underline {Background}$: Let, $ p\in Proj(K[x_0,....,x_n])$ . A double point at $ p\in Proj(K[x_0,....,x_n])$ is the scheme given by the square of the ideal (sheaf) of p. i.e we can consider it ...
1
vote
0answers
38 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, ...
0
votes
0answers
18 views

Tensor product of very ample line bundle with globally generated line bundle is very ample

I think I solved Exercise II 7.5 (d) from Hartshorne's Algebraic Geometry, but I don't know if I used the hypothesis. Those things alway leaves me doubtful if I made a mistake, so I would like to know ...
5
votes
1answer
69 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 $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$, ...
2
votes
1answer
57 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 $\operatorname{deg} x_i = a_i$). ...
0
votes
0answers
15 views

Closed embedding of $\operatorname{Proj} \mathbb C[x][y,z]/(x^2y-xz)$ into $\mathbb P^1_{\mathbb C[x]}$

Consider the graded ring $\mathbb C[x,y,z]$ where $x$ has degree $0$ and $yz$ have degree $1$. Let us denote it by $R[y,z]$ for clarity. Consider the quotient $R[y,z]/(x(xy-z))$. The projection map $R[...
0
votes
1answer
26 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 ...
0
votes
0answers
21 views

Sufficient conditions for a projective curve $X$ over $k$ to satisfy $H^0(X,\mathcal{O}_X) = k$?

Let $X$ be a one-dimensional proper (resp. projective) scheme over a field $k$. By Tag 0BY5 it is necessary that $X$ is Cohen-Macaulay, connected and equidimensional. What are the weakest assumptions ...
3
votes
1answer
48 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 ...
6
votes
1answer
100 views

Hypersurfaces of degree $d$ in $\mathbb{P}^n_k$ that contain a given closed $X$

Let $k$ be an algebraically closed field and consider $\mathbb{P}^n_k$, the be the $n-$dimensional projective space over $k$. It is known that, for any integer $d>0$, there is a bijection between ...
2
votes
0answers
110 views

Automorphism Group of a Variety acts on Local Sections

The motivation/background of my question arises from following thread: Galois morphism - group acting on the variety The original setting is that we have a finite Galois morphism $f: X \to S$, where ...
3
votes
1answer
46 views

Affine charts are dense in projective space

Given a field $k$, we define the scheme-theoretic $n$-th affine space over $k$ by $\mathbb{A}^n_k=\text{Spec}(k[X_1,\dots,X_n])$ and the $n$-th projective space over $k$ by $\mathbb{P}^n_k=\text{Proj}(...
1
vote
0answers
49 views

Why is there a correspondence between $ |\text{Proj}\;S| - V(f) $ and $ \text{Spec}\; (S_{f})_{0}$?

Forgive me for the length of this post, but I feel this question is deserving of some detail. Suppose $ S $ is a positively graded $ A$-algebra, where $ A $ is a ring. That is $ S = \bigoplus_{i=0}^{\...
1
vote
1answer
79 views

Connected Smooth Projective curve $C$ is rational if unirational

Following question: Why and how to see that a connected, smooth, projective curve $C$ (so a so a $1$-dimensional, proper $k$-scheme) is rational if it is unirational. Remark: unirational means that ...
1
vote
0answers
38 views

Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
3
votes
0answers
46 views

Graded global sections of Proj(S) for S a polynomial ring and more general

Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}...
0
votes
0answers
45 views

Differential 1-forms on an irreducible projective variety

Let $k$ be an algebraically closed field and let $X$ be an irreducible projective variety over $k$. I am wondering what the module of differential 1-forms on $X$ is. Since $X$ is a projective variety,...
2
votes
0answers
32 views

Example of family of integral curves with constant gonality and increasing genus

Are there examples of a family of integral curves over some field $k$ which have constant gonality but increasing genus? A related question: If I give you two non-negative integers $n$ and $g$, can ...
0
votes
0answers
17 views

Projective Nullstellensatz and regular rings

If I am not mistaken, and according to the projective Nullstellensatz, we have: $\mathbb{P}_{\mathbb{C}}^n = \mathrm{Proj} (\mathbb{C} [X_0 , \dots, X_n])$, by the correspondence: $ A \to \mathrm{Proj}...
0
votes
0answers
66 views

Maps between projective spaces induced by singular matrices

In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c \times b)$-matrix over $k$. Let $\ell_M: V(b,k)...
0
votes
1answer
43 views

what's the meaning of $B_m/f^n$?

In Page 82 of Qing Liu's book "Algebraic Geometry and Arithmetic Curves", $B$ is a graded ring and $f\in B_+$ is a homogeneous element, it says $B_{(f)}$ is a direct factor of $B_{f}=B_{(f)}\oplus(\...
0
votes
2answers
46 views

Is $D(f)$ the smallest open set of $\operatorname{Spec}B$ such that $D_+(f)\subset D(f)$?

Let $B$ be a graded ring and $\rho:\operatorname{Proj}B\to \operatorname{Spec}B$ the canonical injection, that is, $\forall \mathfrak p\in \operatorname{Proj}B$, $\rho(\mathfrak p)=\mathfrak p$. For ...
1
vote
1answer
77 views

Why don't these morphisms extend?

This is part of exercise 16.5.A in Vakil's notes (http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf). To give an example of why the assumptions in the curve-to-projective extension theorem ...
2
votes
0answers
36 views

Is it possible that irreducible components of curves have greater genus than the original curve?

Let $X$ be a reduced projective curve over a field $k$ with irreducible components $X_1,\ldots,X_r$. Let $p_a(X) = 1- \chi(\mathcal{O}_X)$ denote the arithmetic genus of $X$. Moreover, denote by $X_i'$...
0
votes
0answers
29 views

Restriction of Tautological Sheaf to a Fiber

Let $B$ be a curve (so separated, of finite type, and universally closed $1$-dimensional scheme over a field k) having a closed regular point $b$. Futhermore let $\mathcal{E}$ be a loacally free ...
0
votes
0answers
42 views

Hilbert function

Let $X \subset \mathbb{P}^n(k)$ be a projective variety and $\phi :\mathbb{P}^n(k) \to \mathbb{P}^m(k)$ a morphism with $\deg(\phi_i)=d$. Let $Y$ denote the Zariski-closure of $\phi(X)$. I am trying ...
1
vote
1answer
64 views

Is the Zariski Topology of Proj S the same as the subspace topology of Spec S?

It suffices to show that they have the same set of closed sets. We note the closed sets in Zariski topology of Proj S simply as closed sets, and the ones in subspace topology as induced closed sets. ...
1
vote
1answer
70 views

The universal hyperplane.

In an mathematical article on the net, i find the following paragraph : For a closed scheme $ X \subset \mathbb{P}^n $, there is a natural subscheme of the Grassmannian $ \mathbb{G} ( k , \mathbb{P}^...
4
votes
0answers
106 views

Is a scheme $X$ projective iff every component of $X$ is projective?

Today in a lecture it was claimed Proposition Let $X$ be a proper scheme of finite type and dimension $1$ over a field $k$. Then $X$ is projective. So, I already knew the above statement for ...
1
vote
1answer
107 views

Global sections of the structure sheaf on $\mathbb{P}_{k}^{n}$ in negative degree

This question is about something written on the first page of the lecture notes here. As usual, let $\mathbb{P}_{k}^{n}$ denote the projective space over a field (or indeed any noetherian ring) $k$. ...
0
votes
0answers
57 views

$Proj S$ integral scheme with S not a domain?

My question arises from the comment of this post Projective subschemes and their coordinate rings: "For your first question, it is not true that $\mathrm{Proj} S$ integral implies $S$ is a domain. ...
5
votes
1answer
184 views

Global sections of projective schemes

Let $Y$ be a closed subscheme of $X = \operatorname{Proj} S$, where $S = k[x_0,\dots,x_n]$, $k$ algebraically closed. Then $Y = \operatorname{Proj} (S/I)$ for a homegenous ideal $I$ of $S$. How can we ...
2
votes
2answers
165 views

The vanishing scheme of for a graded ring generated by elements of degree 1 (Vakil 4.5.P)

I am working on the following exercise of Ravi Vakil's Foundations of algebraic geometry. 4.5.P. EXERCISE. If $S_•$ is generated in degree 1, and $f ∈ S_+$ is homogeneous, explain how to define $...
1
vote
0answers
51 views

Properties of the Zariski topology on Proj

Let $S_\bullet$ be a $(\mathbb{Z}_{\geq 0})$-graded ring, $f \in S_+$ be a homogenous element, $I \subseteq S_+$ any homogenous ideal, $V_+(I) := \{p \in ProjS_\bullet | I \subseteq p \}$. I'm ...
2
votes
1answer
46 views

How to choose coordinates for a projective scheme.

Denote by $\mathbb{P}^n:= \mathrm{Proj} \, \mathbb{Z}[X_0, \ldots ,X_n]$. I have read the following somewhere and don't understand it: For a scheme $S$ we choose coordinates $z_i$ with $0 \leq i \leq ...
1
vote
0answers
97 views

Calculating Euler Characteristic of Closed Subscheme

Suppose $X$ is a projective k-scheme of dimension at least one. I want to know how to compute $\chi(X,\mathscr{O}_X(-d))$, the Euler characteristic. My idea was to use the short exact sequence $0 \to ...
1
vote
0answers
37 views

Conditions on $\mathcal{F}$ such that $\chi(\mathcal{F}) = 0$ for a coherent sheaf on a curve over $k$.

Let $X$ be a projective curve (you may also assume Cohen-Macaulayness) over some field $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then the Euler-Poincare characteristic of $\mathcal{...
1
vote
1answer
202 views

Finite morphism $f:X \to \mathbb{P}_k^n$ is surjective?

Let $X$ be an $n$-dimensional projective $k$-scheme and let $f: X \to \mathbb{P}_k^n$ be a finite morphism. Is $f$ necessarily surjective? If not, then what else do we need to impose such ...
0
votes
1answer
87 views

Global section $s$ of ample line bundle such that $X_s$ is everywhere dense

Let $X$ be a projective $k$-scheme of pure dimension $n$ where $k$ is a field. Let $L$ be an ample line bundle on $X$. For all $x \in X$ there is some open neighborhood $U \subseteq X$ and an ...
3
votes
1answer
224 views

Proper curves over some field are projective

I'm looking for a reference of the statement Let $X$ be a proper curve (scheme of dimension one) over the field $k$. Then $X$ is projective. There is a some kind of guided exercise in Liu's ...
0
votes
0answers
55 views

Do torsion-free $\mathcal{O}_X$-modules on curves have dimension one?

Let $X$ be a curve (scheme of dimension 1, say noetherian, projective, Cohen-Macaulay all of whose irreducible components have dimension 1) and let $\mathcal{F}$ be a coherent, torsion-free $\mathcal{...
0
votes
0answers
71 views

Affine schemes which are also projective schemes

Let $A$ be a finite-dimensional $k$-algebra, with $k$ a field. I know (and have also seen it at this site) that $\mathrm{Spec}(A)$ is $k$-isomorphic to a Proj-scheme $\mathrm{Proj}(B) \mapsto \mathrm{...
0
votes
0answers
258 views

The complement of any projective hyperplane is an affine variety

Studying algebraic geometry I'm in trouble with an assumption that my teacher uses: Let $n\geq 1$ and $\mathbb{P}^n$ be the projective n-dimensional space; suppose $H$ is the projective hyperplane in ...
7
votes
1answer
256 views

Behavior of a variety under base change

I am looking for an example of an irreducible variety $X$ say over a field $K$ such that the base change $X_\overline K$ to an algebraic closure is no longer irreducible, and has irreducible ...
2
votes
0answers
87 views

Does a locally projective scheme over an affine scheme embed into projective space?

Definition. A morphism of schemes $f\colon X\to S$ is called locally projective if there is an open covering $S=\bigcup_{j\in J}V_j$, integers $n(j)\geq 0$, and closed embeddings $f^{-1}(V_j)\...
1
vote
0answers
225 views

Why are the global sections of structure sheaf of Proj$S$ just the homogenous elements of $S$?

Let $A$ be a ring and define $S = A[x_{0}, x_{1}, \ldots , x_{r}]$. Let $X = \text{Proj }S$. I would like to show that $\Gamma(X, \mathcal{O}_{X}(n)) = S_{n}$. This is Proposition II 5.13 in ...
1
vote
0answers
178 views

Definition of quasi-projective morphism, and closed immersions and open immersions “commute”

This is a question about another question here. I thought I would make this a separate question since the one there is four years old so I figured it is a long shot trying to get a response, plus this ...
1
vote
0answers
79 views

Understanding the scheme theoreic defintion of the projective plane

The scheme-theoreic definition of projective plane is formed by several steps: 1.Define the line through $(a_1,...,a_n)$ by the closed subscheme of $\Bbb A^{n+1}_C$ defined by $a_jx_i=a_ix_j$. ...
0
votes
2answers
164 views

Scheme-theoreic proof of the projective plane is quasi-compact

The projective space is defined to be $\Bbb P^n(C)=\{\text{line through origin in $C^{n+1}$}\}=C^{n+1}\setminus \{0\}/\sim$ the precise definition coule be found here: https://en.wikipedia.org/wiki/...