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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Configuration scheme of $n$ points

If $X$ is a space, the configuration space of $n$ (distinct) points in $X$ is $C_n(X)=F_n(X)/\Sigma_n$, where $F_n(X) = \{x \in X^n : \forall i,j (i \neq j \Rightarrow x_i \neq x_j)\}$ is the ...
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votes
2answers
181 views

If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's On ...
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1answer
167 views

$X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?

This question came from a proof in Algebraic Geometry by Hartshorne (Chapt3, Corollary 9.6) To be precise, Let $f:X \to Y$ be a flat morphism of schemes of finite type over a field $k$. Then is it ...
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3answers
231 views

Can every element in the stalk be represented by a section in the top space?

Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map?
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71 views

Computing an integral basis of an algebraic function field, $y^4-2zy^2+z^2-z^4-z^3=0$.

I am trying to compute an integral basis for the algebraic extension $K(z,y)$ of $K(z)$ by $y$, with $f(z,y)=0$, $$ f(z,y) = y^4-2zy^2+z^2-z^4-z^3 = 0. $$ $K$ here is either $\mathbb{Q}$ or ...
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1answer
171 views

Pullback of maximal ideal in $k[y]$ is not maximal in $k[x]$.

Let $k$ be a field and let $k[x]=k[x_1,\ldots,k_m]$ and $k[y]=k[y_1,\ldots,y_n]$ be $k$-algebras. Let $\varphi:k[x]\to k[y]$ be a $k$-algebra homomorphism. It can be shown (pretty readily) that ...
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3answers
922 views

Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
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1answer
1k views

Can I go through Hartshorne without knowing much analysis?

I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?
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5answers
314 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
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1answer
154 views

Closure of image of diagonal morphism of S-scheme

Let $X$ be an $S$-scheme with structural morphism given by $f : X \to S$. The image of the diagonal morphism $\Delta : X \to X \times_S X$ is contained in the subset $Z := \{ z \in X \times_S X : ...
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1answer
165 views

Exact sequence in Beauville's “Complex Algebraic Surfaces”

On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent ...
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votes
3answers
334 views

What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
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1answer
165 views

$M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$

Let $S.$ be a graded ring, finitely generated by degree 1 elements as a $S_0$-algebra. Let $M.$ be a finitely generated graded $S.$-module. There exists a natrual map ...
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1answer
69 views

Reconstruct shape of a body from rationality of its projections

There is a closed convex body $S$ in $\mathbb{R}^3$. Areas of its projections on all planes (not only those normal to axes $x,y,z$) are rational numbers. Can we deduce that $S$ is a ball? Replace ...
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3answers
597 views

Localization at a prime ideal is a reduced ring

Here is the question that I came up with, which I am having trouble proving or disproving: Let $A$ be a ring (commutative). Let $p \in Spec(A)$ such that $A_p$ is reduced. Then there exists an open ...
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1answer
205 views

What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
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2answers
697 views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
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4answers
304 views

Complex analysis book for Algebraic Geometers

I know that there exist many questions on this site on complex analysis books but my question is more specific than that. I am looking for recommendations for a concise complex analysis book but with ...
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1answer
165 views

Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
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0answers
433 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
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2answers
479 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the ...
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1answer
204 views

Questions on scheme morphisms

I have some questions on scheme morphisms. I ask pardon for posting them in one thread as they are most likely not worth to be distributed into several threads. Let $X=Spec R$ be a noetherian scheme. ...
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2answers
521 views

Why is the Artin-Rees lemma used here?

I am currently engaged in independent study of algebraic geometry, using Dan Bump's book. One of the exercises in it outlines a proof of the Krull Intersection Theorem, which [here] is the following: ...
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votes
1answer
271 views

Primary decomposition of an ideal (exercise in Reid) [duplicate]

I would like to understand how to use geometry to solve a problem from Reid's book on commutative algebra. The problem is the following Let $k$ be a field and consider the ideal $I = (xy, x - yz) ...
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votes
1answer
226 views

Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?

Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
6
votes
1answer
151 views

Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
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votes
1answer
229 views

Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?

On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$. Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf ...
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1answer
232 views

Construction of the global $\mathbf{Proj}$

I have many questions about the very abstract concept of global $\mathbf{Proj}$. I am following Hartshorne's book Algebraic Geometry, where this concept is on II.7, page 160. Let $(X, ...
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3answers
643 views

Why are projective morphisms closed?

It is a well-known fact that if $X$ is a projective curve and $p \in X$ a smooth point, then any rational map $X \to Y$, $Y$ a projective variety, extends to a rational map $X \to Y$ regular at $p$. ...
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votes
1answer
166 views

Use irreducible fibers to show $X$ is irreducible

Let $\pi:X\rightarrow Y$ be a proper morphism to an irreducible variety and all fibers of $\pi$ are nonempty, irreducible, and of the same dimension. Show $X$ must also be irreducible. Thanks (Any ...
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1answer
625 views

How do I show that this curve has a nonsingular model of genus 1?

Let $C$ be the projective closure of $Z(f) \subset \mathbf{A}^2$ where $f$ is an irreducible polynomial of degree 4 in $x$ and degree 2 in $y$, so $C = Z(f^*) \subset \mathbf{P}^2$ where $f^*$ is the ...
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1answer
384 views

When a scheme theoretical fiber is reduced?

I'd like to ask some basic things in algebraic geometry. Suppose I have a map $\phi:V\to W$, between affine varieties over $k=\mathbb{C}$. for any point $y \in W$. The scheme theoretical fiber is ...
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votes
3answers
556 views

Kähler differentials of affine varieties

I would like to gain some intuition regarding the modules of Kähler differentials $\Omega^j_{A/k}$ of an affine algebra $A$ over a (say - algebraically closed) field $k$. Let us recall the ...
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votes
4answers
76 views

Morphism between two $K$-schemes restricted to an affine subscheme

Suppose that $f:X\longrightarrow Y$ is a morphism between two $K$-schemes. If $U\subseteq X$ is an affine open set, then can we conclude that $f(U)$ is contained is some affine open subset of $Y$? ...
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1answer
94 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
4
votes
1answer
122 views

Hartshorne, exercise II.2.18: a ring morphism is surjective if it induces a homeomorphism into a closed subset, and the sheaf map is surjective

Let $\phi:A\to B$ be a ring morphism, and let $f:X=Spec(B)\to Y=Spec(A)$ be the induced map of affine schemes. I'm trying to show that if $f$ is a homeomorphism onto a closed subset of $Y$ and ...
4
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1answer
82 views

Covering projective variety with open sets $U_i$ such that $\pi^{-1}(U_i) \cong U_i \times \Bbb{A}^1$: How to improve geometric intuition?

I am looking at exercise II 6.3 of Hartshorne. In the first part, he asks to show the following. If $V \subseteq \Bbb{P}^n$ is a projective variety (over some field $k$), let $X = C(V)$ denote its ...
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votes
1answer
129 views

Questions about the definition of a Cartier Divisor from Liu pg 256

I am reading Definition 1.17 of Liu on what a Cartier Divisor is: I have several questions concerning this definition. Question 1: It makes sense to say to me that an element $D \in ...
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votes
1answer
92 views

Etale spaces of a presfeaf and the associated sheaf

Given a presheaf $\mathcal{F}$on a topological space $X$, one can construct the etale space $\pi_1 : Y_1\to X$. Let us now look at the associated sheaf $\mathcal{F}^+$ as a presheaf and construct the ...
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2answers
198 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
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2answers
122 views

Does the fibres being equal dimensional imply flatness?

Let $f: Y \to X$ be a morphism of varieties (proper if necessary). I read from a paper that if all the fibres of $f$ are of the same dimension then $f$ is flat. This seems skeptical for me, and I ...
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1answer
438 views

Projective Normality

What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
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1answer
188 views

“Push-Pull” Morphisms of Higher Direct Image Sheaves

(This is 20.7.B in Ravi Vakil's notes) Suppose $f:Y \to Z$ is any morphism, and $\pi: X\to Y$ is quasicompact and quasiseparated. Suppose $\mathcal{F}$ is a quasicoherent sheaf on $X$. Let ...
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2answers
135 views

$\operatorname{Spec}A$ is a finite set if and only if $A$ is a finite dimensional vector space

Let $k$ be an algebraically closed field, and $A$ be a finitely generated $k$-algebra with no nilpotents. Show that $\operatorname{Spec}A$ is a finite set if and only if $A$ is a finite dimensional ...
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1answer
54 views

Question on geometrically reduced, geometrically connected.

I have a question from a book which I am trying to attempt. Let $k$ be a field not of characteristic 2 and let $a\in k$ be not a square (i.e. for all $b\in k$, $b^{2}\neq a$). I want to show that ...
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1answer
115 views

Scheme glued out of finitely many spectra of local rings

Let $X$ be a noetherian separated scheme over the spectrum of a field $K$. By definition, I can find an open covering of $X$ by finitely many affine schemes $Spec(R_i)$ where each $R_i$ is a ...
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votes
2answers
296 views

Sheafs and closed immersion

Let $f:X \rightarrow Y$ be a continuous map of topological spaces, such that it is closed immersion. Let $\mathfrak{F}$ and $\mathfrak{G}$ be sheafs on $X$ and $Y$ respectively. How to show, that ...
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votes
3answers
119 views

Why isn't the associated sheaf transformation onto?

What is an example of a presheaf $F$ such that the usual morphism $F\to \tilde F$ to the associated sheaf is not onto on all sections, i.e. there exists an $X$ and $F(X)\to \tilde F(X)$ is not onto?
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2answers
127 views

Did I write the right “expressions”?

$9$. Consider the parametric curve $K\subset R^3$ given by $$x = (2 + \cos(2s)) \cos(3s)$$ $$y = (2 + \cos(2s)) \sin(3s)$$ $$z = \sin(2s)$$ a) Express the equations of K as polynomial ...
1
vote
1answer
235 views

Birational map from a variety to projective line

This is exercise $4.4$ part (c) of Hartshorne's book. Let $Y$ be the nodal cubic curve $y^{2}z=x^{2}(x+z)$ in $\mathbb{P}^{2}$. Show that the projection $f$ from the point $(0,0,1)$ to the line $z=0$ ...