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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12 views

Tropical top self-intersection numbers of boundary divisors in toroidal embeddings

Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
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24 views

A specific notation for invertible sheaf

I am reading Milne's Jacobian Varieties, where on the discussion after Proposition 2.1 in Chapter 2, he says the following: Let $P \in C(k)$, where $C$ is a proper nonsingular curve over $k$ and ...
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29 views

Comparing covers

Considering the Zariski topology, let $$V = \bigcup_{i \in I} U_i$$ be a maximal open cover of $V$ by basic open sets. Similarly, let $$V' = \bigcup_{j \in J} W_j$$ by the maximal open cover of $V'$ ...
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38 views

If we have a sheaf of abelian groups, how do we know that the restriction maps commute with our addition?

Let $X$ be a topological space and $F,G$ be presheaves of abelian groups on $X$. Then given two presheaf morphisms $\phi,\psi:F\to G$, we'd like to define the sum of the two morphisms, by taking (for $...
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53 views

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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198 views

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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30 views

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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45 views

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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68 views

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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52 views

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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133 views

Where to read about sheaves?

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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37 views

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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49 views

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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41 views

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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80 views

de Rham cohomology of singular varieties: why completion?

If $X$ is a smooth variety over an algebraically closed field $k$ of characteristic zero one can define algebraic de Rham complex $$ \mathcal{O}_X \to \Omega^1_X \to \ldots \to \Omega_X^n, $$ where $\...
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34 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 ...
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1answer
30 views

Concyclicity of $4$ points using algebraic geometry

Consider the line $L_1:ax+4y-1=0$ and a circle $S:x^2+y^2-10x+2y+10=0$. The line intersects the circle at $2$ distinct points $A$ and $B$. Another line $5x-12y-67=0$ intersects the circle $x^2+y^2+6x+...
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32 views

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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2answers
62 views

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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87 views

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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68 views

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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32 views

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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47 views

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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67 views

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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51 views

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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1answer
53 views

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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43 views

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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1answer
56 views

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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41 views

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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20 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 ...
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24 views

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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16 views

Periods and Dual-Periods of Riemann Surface?

I'm unclear on to what extent the periods and dual-periods of a Riemann surface determine the complex structure of the surface. Perhaps to take a nice example, I'll consider a hyperelliptic curve $y^{...
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190 views

A celestial topology?

I recently asked for natural topologies on the set of lines in $\mathbb R^2$. Now I'm aiming for a similar question on the set $S_p$ of conic sections in $\mathbb R^2$ sharing the same focus $p$ (but ...
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1answer
88 views

Spectrum in functional-analysis and algebraic geometry

Why do we use the notion "spectrum" both in functional-analysis and in algebraic geometry? Are there any analogies?
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38 views

Projection from a point on a curve

Let $k$ be an algebraically closed field, char $k=0$, and let $C\subset\mathbb{P}_k^2$ be a nonsingular projective plane curve of degree $d$. Let $O\in C$, $L\subset\mathbb{P}_k^2$ a line not ...
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1answer
52 views

Veronese Variety

Can anyone explain proposition 5.1 in these notes: http://www.math.utah.edu/~bertram/6140/Examples.pdf What is the general map between the Veronese Variety and the image of the veronese map? ...
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41 views

Question about the lattice of the O'Grady 6-dimensional example

This is perhaps a silly question, but I could not find neither a reference nor an answer by myself. Let $X=\widetilde{M}_{v}$ be the $6$-dimensional irreducible holomorphic symplectic manifold ...
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1answer
149 views

Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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34 views

Is there any product formula for local zeta function?

Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the ...
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92 views

Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
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83 views

A scheme is affine iff the natural map $X\to \operatorname{Spec}\Gamma(X)$ is an isomorphism

We know that the functor $\operatorname{Spec}: \mathsf{Rings}^{\text{op}}\to \mathsf{Schemes}$ is right adjoint to the global section functor $\Gamma: \mathsf{Schemes}\to \mathsf{Rings}^{\text{op}}$. ...
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69 views

When are powers of prime ideals primary?

This is a follow up to: Normal domains and powers of height one primes In the comments to the linked question, user26857 noted that the prime ideal $P = (x,z)$ in the Noetherian normal domain $k[x,y,...
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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 ...
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1answer
40 views

Silverman AEC 11b

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 ...
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36 views

Example of projection map having non-reduced fibers

This question stems from Oliver Debarre's Higher-Dimensional Algebraic Geometry, proposition 5.7. Let $X$ be a normal quasi-projective variety over an algebraically closed field of characteristic $p &...
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1answer
45 views

If $X(F) \cap Y$ is dense in $Y$, then $Y$ is defined over $F$.

Let $k$ be an algebraically closed field, $F$ a subfield of $k$, $A$ a finitely generated, reduced $k$-algebra, and $A_0$ an $F$-subalgebra of $A$, of finite type over $F$, such that the canonical $k$-...
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1answer
28 views

Fibred product of sets $X\times_Z Y=\{(x,y)\in X\times Y: \alpha(x)=\beta(y)\}$ satisfies the universal property.

This is Exercise 1.3.N from Vakil's notes of Algebraic Geometry. The following is the diagram defining the universal property of fibred product: Show that in $\mathit{Sets}$, $$X\times_Z Y=\{(x,...
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32 views

Global Sections of a particular projective scheme

Let $A=k[\![x]\!]$ and consider the closed subscheme $X=V_{+}(xy_2) \subseteq \mathbb{P}_A^1$ (where I write $(y_1:y_2)$ for the homogeneous coordinates in $\mathbb{P}^1$). I am confused about how the ...