# Questions tagged [projective-schemes]

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

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### Does the spectrum of the stalk of a point of a noetherian scheme contain an open subscheme about that point?

I am just proving the local consistency of Hilbert polynomials on flat families. All the proofs I have found till now, they use a map $i_{y}: \operatorname{Spec}(\mathscr O_{Y,y})\rightarrow Y$ and ...
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### Showing that $\mathbb P^n_A\rightarrow \operatorname{Spec}A$ is proper.

$\newcommand{Proj}{\operatorname{Proj}}\newcommand{Spec}{\operatorname{Spec}}\newcommand{\p}{\mathfrak p}\newcommand{\P}{\mathbb P}\newcommand{\Z}{\mathbb Z}$ Ok, so I understand what we are doing in ...
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### Different Notions of Degree Agree

Vakil's (late February 2024 edition) notes define the degree of a line bundle $\mathscr{L}$ on projective curve $C$ over field $k$ to be $\chi(C, \mathscr{L}) - \chi(C, \mathscr{O}_C)$. On the other ...
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### Quotient of Graded Rings $S \to S/f$ inducing Homeomorphism on Proj

Let $A$ be a Noetherian ring, and let $X$ be a closed subscheme of of $\mathbb{P}^r_A$. We define the homogeneous coordinate ring $S:=S(X)$ of $X$ for the given embedding to be $A[x_0, ..., x_r]/I$ (...
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### Coordinate Ring of the Cone of an Integral Projective Scheme a Domain (Generalization of Hartshorne's Ex II.5.14)

I have several questions around a part of Exercise 2.5.17(a) from Hartshorne's Algebraic Geometry (p 126). The setting: Let $k$ be a field, and let $X$ be a closed subscheme of of $\mathbb{P}^r_k$. We ...
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### A point inside a projective variety gives a (quasi) projective morphism

Let $X\to \operatorname{Spec } K$ be a projective integral variety. Let $L|K$ be a field extension and consider an $L$ point $x\colon \operatorname{Spec} L\to X$ in $X$ (morphism over $K$). Under ...
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### Structure of a scheme on $\operatorname{Proj} A$

Let $A$ be $\mathbb Z_{\geq 0}$ graded commutative ring. Now, in almost every resource I read, we define the structure a scheme on $\operatorname{Proj} A$, by taking the distinguished basis $D_+(f)$ ...
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### What went wrong computing the normal bundle of a section in projective bundle?

Suppose $X$ is a scheme (possibly smooth / $\mathbb C$), $E$ is a vector bundle of rank $e$ on $X$ with corresponding projective bundle $p:\mathbb PE \to X$, and $s: X \to \mathbb PE$ is a section. I ...
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### On the homeomorphism $U_f\subset \operatorname{Proj}A\leftrightarrow \operatorname{Spec}(A_f)_0$

Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this ...
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### On the bijection between homogenous prime ideals of $A_f$ and prime ideals of $(A_f)_0$

Let $A$ be a $\mathbb{Z}^{\geq0}$ graded ring, and $f$ an element of positive degree. It is well known that the homogenous prime ideals of $A_f$ are in bijection with prime ideals of $(A_f)_0$, that ...
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### Inverse image of a $0$-dimensional subscheme under a finite morphism

Let $f\colon X\rightarrow Y$ be a finite surjective morphism of complex projective varieties of the same dimension. We may suppose $Y$ is smooth. Let $Z\subset Y$ be a $0$-dimensional closed subscheme ...
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### Reduced, irreducible and integral schemes

I am studying properties of schemes and I have the following problem. Let $R$ be a ring, $S=Spec(R)$, $n\in\mathbb{N}$ an integer. Show that the following are equivalent: $S$ is reduced (resp ...
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### [Cor 5.3.24 in Qing Liu]If $X$ is a projective scheme over a DVR $A$, then any coherent sheaf of ideal of $\mathcal{O}_X$ is flat over $A$?

The problem comes from the proof of Corollary 5.3.24 in Qing Liu's book Algebraic Geometry and Arithmetic curve. Let $\DeclareMathOperator{\Spec}{Spec}S=\Spec A$ be the spectrum a discrete valuation ...
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### global section of a B-scheme defines a morphism to projective $B$-scheme

This is problem 7.3.O in Vakil's AG notes (July 31, 2023 version). I am not sure how to define the map locally. Problem 7.3.O. Let $B$ be a ring and $X$ a $B$-scheme. Suppose $f_0,\cdots,f_n$ are n+1 ...
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### Example of a family of ample divisors $\{A_m\}_{m\geq 1}$ on a smooth projective variety $X$ such that $mA_m$ has a basepoint?

I know this famous example due to Kollár: Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover ...
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### Vanishing ideal of a subset of $\operatorname{Proj} S_\bullet$

I am reading FOAG by Vakil. In 4.5.K, I am asked to define $I(Z)$ where $Z\subseteq\operatorname{Proj}S_\bullet$ and $S_\bullet$ is a graded ring. The immediate idea of mine is to define it as I(Z):=...
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### If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities,...
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