The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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3
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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 ...
2
votes
1answer
161 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
1
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1answer
67 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 ...
13
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3answers
589 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 ...
11
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2answers
678 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 ...
10
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1answer
186 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?
9
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4answers
291 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 ...
8
votes
1answer
161 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 ...
7
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0answers
425 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 ...
7
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1answer
203 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. ...
7
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2answers
515 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: ...
6
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1answer
211 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
226 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 ...
6
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1answer
227 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, ...
6
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3answers
634 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 $p$. ...
6
votes
2answers
461 views

Elementary question about Cayley Hamilton theorem and Zariski topology

A question about a proof of Cayley's Hamilton theorem using Zariski topology. "The set $C$ of all matrices of size $n \times n$ (over an algebraically closed field $k$) with distinct eigenvalues is ...
5
votes
1answer
163 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 ...
5
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1answer
615 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 ...
5
votes
1answer
147 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 ...
5
votes
1answer
370 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 ...
5
votes
3answers
550 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 ...
4
votes
4answers
71 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 $Y$? ...
4
votes
1answer
93 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
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1answer
121 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
80 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 ...
4
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1answer
126 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 ...
4
votes
1answer
89 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 ...
3
votes
2answers
194 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 ...
3
votes
2answers
112 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 ...
3
votes
1answer
416 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 ?
3
votes
1answer
182 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 ...
2
votes
2answers
134 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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
2answers
285 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 ...
2
votes
3answers
116 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?
2
votes
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
228 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 $z=0$ ...
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votes
1answer
135 views

Irreducible polynomials and affine variety

Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
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2answers
208 views

Are sets given in parametric form always algebraic?

If a set is given in parametric form by polynomials, is this set always closed (Zariski topology), i.e algebraic? For example, take $X=\{(t,t^{2},t^{3}): t \in \mathbb{A}^{1}\}$ and ...
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3answers
275 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
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524 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
8
votes
1answer
191 views

Tangent sheaf of a (specific) nodal curve

Given a nodal (= reduced, connected, projective, having only ordinary double points as singularities) curve $C$ consisting of 5 $\mathbb P^1$ labeled $C_0, D_1, D_2, D_3, D_4$ such that $D_i$ ...
8
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1answer
162 views

Trying to parse a definition in Silverman's EC book

Let $C_1\subset \mathbb{P}^{N_1}$ and $C_2\subset \mathbb{P}^{N_2}$ be two curves. Then a map $\phi:C_1\to C_2$ can be defined as $$\phi=[f_0,\ldots,f_{N_2}],$$ where each $f_i$ is a homogeneous ...
7
votes
2answers
110 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
7
votes
1answer
339 views

Deligne's formula

Let $M$ be some $A$-module and $f \in A$. Why do we have an isomorphism $$\varinjlim_n \hom_A(f^n A,M) \cong M_f \text{ ?}$$ Background. Let $X$ be a scheme, $U$ an open subscheme, and $F,G$ ...
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votes
2answers
205 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
7
votes
2answers
260 views

Principal maximal ideals in coordinate ring of an elliptic curve

Let $E$ be an elliptic curve over an algebraically closed field, and let $R$ be the coordinate ring of $E \setminus \{\infty\}$. I have read somewhere that $R$ has no principal maximal ideal. But I ...
7
votes
1answer
264 views

why the K3 surfaces are minimal surfaces

I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read beauville's book on complex algebraic surfaces: ...
7
votes
1answer
252 views

Is the dualizing functor $\mathcal{Hom}( \cdot, \mathcal{O}_{X})$ exact?

In Hartshorne's Algebraic Geometry II.8.20.1 (page 182), he takes the dual of Euler sequence $$0 \rightarrow \Omega_{X/k} \rightarrow \mathcal{O}_{X}(-1)^{n+1} \rightarrow \mathcal{O}_{X} \rightarrow ...