# Tagged Questions

The tag has no usage guidance.

177 views

### How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed ...
36 views

### The $\mathrm{Proj}$-construction and inverse limits

I have a couple of questions about existence of certain inverse limits in the category of schemes (I am also happy about links to relevant literature... in the stacksproject I only found the affine ...
21 views

### Existence of scheme quotient

I have a morphism of schemes $X\to S$ which is very nice: flat, proper, finitely presented. I also have a finite group $G$ acting (faithfully, but not necessarily freely) on $X/S$. 1) I am quite ...
29 views

### Sheafifying direct sum of twists

Let be $X\subseteq \mathbf{P}^r$ a smooth projective variety and let be $\mathscr E$ an invertible sheaf over $X$. Let $$M=\bigoplus_{n\geq 0} H^0(\mathscr E(n))$$ as a module over the polynomial ...
110 views

### Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ...
41 views

### Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
129 views

45 views

### Normal projective varieties and its coordinate ring

Let $k[X_0,...,X_n]$ be a polynomial ring over an algebraically closed field of characteristic zero and $I$ an ideal of $k[X_0,...,X_n]$ generated by homogenous polynomials. Denote by $X$ the ...
79 views

### Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
135 views

### Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
101 views

### Exercise in R.Vakil 18.4.L: Ample line bundles and finite morphism

Here's the question: R.Vakil, Exercise 18.4.L: Suppose $\mathcal{L}$ is a base-point free invertible sheaf on a proper variety $X$, and hence induces some morphism $\phi: X\rightarrow\mathbb{P}^n$....
I need help to solve the following problem that appears on page 55 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $R = k[X,Y,Z]$, $F \in R$ an irreducible form of degree $... 0answers 34 views ### Connection between Chladni Plates and Projective Geometry? Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be explained/... 2answers 168 views ### Perfect complexes and the derived category of a smooth projective variety I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in$D^b(X)$for$X$smooth projective is then ... 1answer 122 views ### Rational maps from noetherian$k$-scheme to projective$k$-scheme extend over regular codimension 1 sets I am trying to solve and am currently stuck on Vakil's problem 16.5.B which is asked as follows: Suppose$X$is a Noetherian$k$-scheme and$Z$is a irreducible codimesion$1$subvariety whose ... 1answer 85 views ### How to see$\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$as a ring? Let$S$be a finitely generated graded$A$-algebra. For each homogeneous$f\in S_+$, we have a scheme structure$D(f)\cong \operatorname{Spec} S_{(f)}$where$S_{(f)}$denotes the zeroth piece of the ... 2answers 149 views ### Understanding Hartshorne's proof that every projective morphism is proper. In chapter$II$of Hartshorne, theorem$4.9$shows that every projective morphism is proper, using the valuative criterion for properness. I understand how the required morphism is constructed, but I ... 1answer 214 views ### Blow-up and base change Consider a complex smooth (projective) surface$X$and a blow-up$\epsilon:S\longrightarrow X$at a point$x\in X$. Let$\sigma\in\text{Aut}(\mathbb C)$be a field automorphism and moreover let $$\... 1answer 67 views ### “Twist” of \mathbb P^n_K through a field automorphism. This question is closely related to this recent one. Suppose that s:X\longrightarrow\text{Spec}\, K is a variety over K (i.e. a K scheme, separated, proper and geometrically integral) and ... 1answer 156 views ### The importance of the structural morphism of a projective variety. In scheme-theory, "The projective n-dimensional space over k" is defined as \mathbb P^n_k:=\text{Proj}(k[T_0,\ldots,T_n]). Moreover \mathbb P^n_k is endowed with a structure of variery over k... 2answers 81 views ### Structure sheaf of Proj \ S in terms of compatible stalks Let S be a graded ring. I was wondering if someone could please explain me how I can interpret structure sheaf of Proj \ S in terms of compatible stalks? Thank you! Edit: This is Exercise 4.5.M. ... 1answer 81 views ### What is the canonical morphism of \mathbb{P}^n_A to \text{Spec }A? On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme \mathbb{P}^n_A, for A a ring, as Proj (B), for B=A[x_0,\dots,x_n]. Later, he ... 1answer 166 views ### Compute the cohomology of projective schemes In Hartshorne's book, Section 3.5, the cohomology of projective spaces is computed. How to compute the cohomology of projective schemes? Maybe the general case is complicated, please look at the ... 0answers 41 views ### What is the definition of I(Z) when Z \subseteq Proj S Let S be a graded ring. In the notes of Ravi Vakil, he only says "define I(Z) \subseteq S_+", when Z \subseteq Proj S, where S_+ is all the elements in S that are sum of positive degree ... 0answers 62 views ### Point at infinity of the scheme \mathbb{P}^1 Let \mathbb{P}^1_k be the scheme defined as either \text{Proj }k[t] or obtained by gluing two affine lines appropriately. What is the point at infinity which in topology usually is given by [0:1]... 1answer 58 views ### Confused about short exact sequence involving \mathcal{O}_{\mathbb{P}^n} and \mathcal{O}_Z for \pi:Z\rightarrow \mathbb{P}^n closed embedding If \pi: Z\rightarrow \mathbb{P}^n is a closed embedding where Z is the zero set of some degree homogeneous polynomial, we have:$$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow \mathcal{O}... 1answer 55 views ###$(-1)$-curves and base change along a field automorphism. Suppose that$E\subseteq S$is a$(-1)$-curve inside a non-singular complex projective surface. By a$(-1)$-curve$E$, I mean that$E\cong\mathbb P^1_\mathbb C$and$E^2=-1$. Now consider a field ... 1answer 87 views ### Does the canonical morphism commute with direct image functor? I am trying to prove the representability of the Quotient functor. I have the following problem. Let$\phi \colon T \to S$be a morphism of noetherian schemes and let$F$be a coherent sheaf on$\...
In Hartshorne's book Algebraic Geometry in Chapter III Section 12 we have the following situation: $f\colon X\to Y$ is a projective morphism of schemes, $Y$ is the affine spectrume of a ring $A$ , ...