# Questions tagged [schemes]

The concept of scheme mimics the concept of manifold obtained by gluing pieces isomorphic to open balls, but with different "basic" gluing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, the spectrum of a commutative ring with unit.

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### How to show $\operatorname{Pic}(X)=0$? Exercise $14.2$.Q Vakil's notes

I'm reading Vakil's notes and I'm struggling with the exercise $14.2$.Q. I've been able to prove everything except $\operatorname{Pic}(X)=0$ with $$X=\operatorname{Spec}\frac{k[x,y,z]}{(xy-z^2)}.$$ ...
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### What is a generically reduced scheme?

I am reading the book "3264 & All That Intersection Theory in Algebraic Geometry". In the following definition (see page 30) Definition 1.22. Let $f:Y\rightarrow X$ be a morphism of ...
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### Quasi-separatedness of diagonal of DM stacks and group stabilizer of a point

I was studying Jarod Alper's book on stacks (https://sites.math.washington.edu/~jarod/moduli.pdf) and after the definition of stacks and some basic discussion about them, he proposes the following ...
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### deformation to the normal cone in homotopy purity

I'm reading the proof of homotopy purity theorem. A key step in the proof is \textbf{Deformation to the normal cone}: Let $Z\xrightarrow{i}X$ be a closed immersion in $Sm/S$, where $S$ is noetherian ...
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1 vote
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### Transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$

I am trying to understand the transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$. I've been stuck on this while reading these notes (Proposition 1.8, page 3) on divisors and invertible ...
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### Surjective étale morphism between normal schemes

(1) If I have $X,Y$ noetherian schemes, locally of finite type over a field $k$, normal and connected, can I somehow conclude that a surjective étale morphism $X \to Y$ is finite? I have seen some ...
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1 vote
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### Counterexample to one version of Chevalley's theorem?

Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, then $f(C)$ is constructible. Here, constructible means $C$ is ...
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### Is there a hypothesis missing in this statement of the See-Saw principle?

In Huybrecht's 'Fourier-Mukai Transforms in Algebraic Geometry', the see-saw principle is stated in Proposition 9.4 as follows. Let $X$ be an irreducible complete variety over a field $k$, $T$ an ...
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### Describing all endomorphisms of elliptic curves

Let $k$ be an algebraically closed field. A curve is a separated integral $k$-scheme of finite type over $k$ and of dimension one. An elliptic curve $E$ is a smooth projective curve of genus one (a $k$...
1 vote
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### Increasing sequence of reduced closed subschemes.

Let $k$ be a perfect field, $V$ be an irreducible smooth $k$-scheme and $W$ is an open dense subset of $V$. Let $F$ be the complement of $W$. In this lemma, Morel states that there is an increasing ...
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### Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S =$ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
1 vote
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### Existence of a model for $(X,L)$

Recently I was reading Moriwaki's work about the heights on arithmetic varieties, and I came across the following doubt about the setting. Let $(X,L)$ be a couple where $X$ is a projective nonsingular ...
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### Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$ for any $Z$ is a consensus which I agree with, but then regarding the hilbert ...
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### Can $\nu_Y (f) = 0$ for every prime divisor containing $Z = \overline{\{z\}}$, where $f \notin \mathcal{O}_z$?

This is about the argument in Hartshorne exercise III.6.8(a), which is supposed to show $X_s$ form a base for the topology of a noetherian, integral, separated, locally factorial scheme $X$, where $s$ ...
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### Where is the error in this proof that all morphisms of schemes are quasi-compact?

Lemma 1. All preimages of affine open subschemes are affine (Inverse of open affine subscheme is affine). Lemma 2. All affine schemes are quasi-compact (Proof that an affine scheme is quasi compact). ...
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### Confusion regarding definition of Grassmannian

I had been thinking of Grass$(p,r)$ as the set of $r$ dim subspaces of $k^p$ where $k$ is whatever ground field we take in the context. We then topologize this set. Now in one of his papers, Nitsure ...
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1 vote
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### Global functions on arithmetic varieties

Let $f:X\to\operatorname{Spec} O_K$ be an arithmetic variety where $K$ is a number field and $O_K$ is its rings of integers. We assume that $X$ is integral, projective, regular and $f$ is flat. If ...
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### Hartshorne II.3.10

I'm working on Exercise II.3.10 from Hartshorne and I'm baffled by what should be a relatively simple exercise on schemes. The exercise states If $f:X\to Y$ is a morphism (of schemes), $y\in Y$ a ...
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### pullback of schemes

So we had the following statment in the lecture: I just dont get why this equation is true, even though it probably just follows immediately from the universal property of the pullback. Thanks in ...
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### Semicubical parabola is not isomorphic to the affine line (module of differentials)

Here is exercise 1 from chapter 8.1 of Bosch - Algebraic Geometry and Commutative Algebra: (Exercise:) For a field $K$, consider the coordinate ring $A = K[t_1, t_2]/(t_2^2 − t_1^3)$ of Neile’s ...
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### Constants of universal derivation on an $R$-algebra $A$ under localization ($d: S^{-1}A \to S^{-1}\Omega_{A/R}$)
Suppose that $A$ is an $R$-algebra, with both $R, A$ integral domains. Let $B = S^{-1}A$ be a localization of $A$. Let $$\partial_A: A \to M$$ $$\partial_B: B \to S^{-1}M$$ be an $R$-derivation on $A$ ...
### Map from X to $\mathbb P^1_k$ of degree $3$ with $X$ curve of genus $3$.
Let $X$ be a curve over $k$ algebraically closed field. Here $X$ is a proper, smooth, connected, $\text{dim}(X)=1$ scheme over $k$. Suppose that $g(X) = 3$ with \$g(X) = \text{dim}_k H^1(X,\mathcal O_X)...