# Tagged Questions

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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### the ideal defining a tangent space is radical?

In algebraic geometry, if $X$ is an affine variety than the tangent space $T_pX$ of $X$ at its point $p$ is the affine variety defined as the set of zeros of the ideal $J$ generated by the linear ...
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### Is a fiber product of flat morphisms flat?

Suppose we have morphisms of schemes $f : X\rightarrow S$ and $g : Y\rightarrow S$, and a morphism $Z\rightarrow X\times_S Y$ such that the induced morphisms $Z\rightarrow X, Z\rightarrow Y$ are flat. ...
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### Is the projective line minus one point always isomorphic to the affine space?

I'm thinking about the following problem: If I take a general point $p \in \mathbb{P}^1$ out of the projective line, is $\mathbb{P}^1 - \{ p \}$ isomorphic to the affine space $\mathbb{A}^1$? I ask ...
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### Maximal unramified intermediate extension of DVRs

Let $A\rightarrow B$ be a finite tamely ramified extension of discrete valuation rings. Does there exist a DVR $C$ such that $A\subseteq C\subseteq B$ with $C$ unramified over $A$ and $B$ totally ...
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### Inverse limit of Hopfalgebras

My Question relates to Corollary 2.7 of http://www.jmilne.org/math/xnotes/tc.pdf So Let $k$ be a field and $\mathbb{G}_i$ be an projective system of affine $k$-groupschemes. I want to know if the ...
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### Why do we consider scheme-theoretic images?

Solving Problem 3.2.3 of Qing Liu, If $f:X\to Y$ is a quasi-compact immersion, then $f$ can be decomposed by $i\circ u$ where $i$: closed immersion, $u$: open immersion. If we can reduce this ...
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### Regular points on an effective Cartier divisor is regular on the whole scheme?

I think this should be an easy question to answer but I'm being unable to prove it. Vakil, in one of his notes, states that Suppose $X$ is a finite type $k$-scheme (such as a variety), and $D$ is ...
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I'm working through Mumford's Red Book, and after introducing the definition of a sheaf, he says "Sheaves are almost standard nowadays, and we will not develop their properties in detail." So I guess ...
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### primes of the strict henselization

I'm trying to get some intuition for the (strict) henselization of a local ring. Let $A$ be a local ring with maximal ideal $m$. I'm happy to assume it is Noetherian and normal. Let $p\subset m$ be a ...
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### Normal bundle of a point

Let $X$ be a projective variety over a field $k$. I am trying to understand the notion of the normal bundle of a closed immersion. Let $x$ be a closed point of $X$. What is the normal bundle of $x$ ...
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### Intersections of algebraic varieties

I was looking at the question here, and I wasn't sure why it was obvious that $V\left(\sum\limits_\lambda I_{\lambda}\right) \subseteq \bigcap\limits_\lambda V(I_{\lambda}).$ But in typing my follow-...
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### What does Hilbert series of Monomial ideal describe?

I am trying to understand the point of hilbert series of monomial ideals. I am confused because Macaulay has commands for hilbertSeries, hilbertPolynomial and hilbertFunction. What does Hilbert ...
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### Does taking $\mathbb{C}$-points of a scheme preserve pullbacks or pushouts?

Let $K$ be the field $\mathbb{C}$ of complex numbers and let $X$ be a scheme of finite type over $S=\operatorname{Spec(K)}$. The set $X(K)=\hom_{Sch/S}(S, X)$ of $K$-rational points of $X$ carries a ...
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### Finding the normalization of $K[X,Y]/(Y^5-X^7-XY^5)$

I understand that for "relatively simple" cases, we can compute the normalization of a coordinate ring, such as $K[X,Y]/(Y^2-X^3)$, quite easily (consider $Y/X$). However, how would one approach ...
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### cohomology of total space

Suppose $\mu:E\to B$ is a fiber bundle with fiber $F$. Furthermore, $F$ and $B$ have vanishing odd dimensional cohomology group. Is it true that $E$ has vanishing odd dimensional cohomology group? You ...
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### What is an $S$-morphism?

Let $S$ be a scheme, let $X$ be a reduced scheme over $S$, and let $Y$ be a separated scheme over $S$. Let $f$ and $g$ be two $S$-morphism of $X$ to $Y$ which agree on an open dense subset of $X$. ...
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### Is a smooth ring a domain?

I know that smooth and regular is "quasi" the same and that a regular local ring is a domain. Here I start with the definition $A \to B$ is smooth if and only if for every square zero extension of $A$-...
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### How to determine redundant elliptic curves?

When we enumerate elliptic curves $y^2 = x^3 + ax + b$ over a finite field, how do we determine redundant ones, i.e. ones that are equivalent to others?
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### Confused about how to work with local rings for an arbitrary variety

Time for my latest dumb algebraic geometry question as I try to self learn through Hartshorne. I am currently trying to do the exercises in chapter 1 without looking at solutions. One method I have ...
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### Families as fibres of a morphism

In both Algebraic Geometry by Hartshorne and Geometry of Schemes by Eisenbud and Harris, the authors describe the notion of a family of schemes as being the fibres of a morphism $f:X\to Y$. Or as ...
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### nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
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### Abstract algebraic definition of dual tangent spaces

I know that if $(M,\mathcal{A})$ is a smooth manifold, the dual tangent space at $p\in M$ can be defined as $$T^*_pM=I_p/I_p^2,$$ where $I_p$ is the ideal of the ring $C^\infty(M)$ consisting of ...
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### When is the contraction of a maximal ideal still maximal?

Let $k$ be an algebraically closed field, $F$ a subfield of $k$, $A$ a reduced, finitely generated $k$-algebra, and $A_0$ a finitely generated $F$-subalgebra of $A$ for which $A = k \otimes_F A_0$. ...
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### 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 ...
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### Let $f:X\to Y, g:Y\to Z$ be regular maps of quasi-projective varieties, each with dense image. Then $g\circ f$ has dense image.

I'm fairly new to algebraic geometry, so I may be overlooking a very simple solution, but at the moment, I don't see how to prove this. If $X$, $Y$, and $Z$ were affine varieties, then I would be able ...
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### elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For ...
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### What's a morphism? Well it's a morphism.

I'm confused on the definition of an "$F$-morphism" of $F$-varieties. The textbook is Springer, Linear Algebraic Groups. Let $k$ be an algebraically closed field, and $F$ a subfield of $k$. The ...
Some search on the internet and this site didn't result in any topic about this question of Silverman's The Arithmetic of Elliptic Curves: Let $W \subset \mathbb{P^n}$ be a smooth algebraic set, each ...