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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8
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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 ...
1
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0answers
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 ...
4
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2answers
93 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.
3
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2answers
84 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 $\Gamma: \mathsf{Schemes}\to \mathsf{Rings}^{\text{op}}$. ...
1
vote
1answer
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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0answers
35 views

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 ...
5
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0answers
72 views

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 ...
1
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1answer
42 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 ...
1
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0answers
38 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 &...
0
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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 $k$-...
1
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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,...
0
votes
1answer
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 ...
0
votes
0answers
61 views

Normal domains and powers of height one primes

Let $A$ be a Noetherian normal domain, and $P$ a height 1 prime of $A$. Then the localization $A_P$ is a discrete valuation ring. Certainly $PA_P\cap A = P$. Does this also hold for higher powers of $...
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0answers
39 views

Special cases of prime avoidance theorem

Let $\{p_i\}$ be a family of minimal prime ideals in a commutative ring $R$ with $1$, and let $I$ be a finitely generated ideal of $R$ such that $I\subseteq \cup p_i$. Can we deduce that there exist $...
1
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0answers
26 views

Why is $(\textrm{Ker } \beta)^0 = (\textrm{Ker } \alpha)^0$

Let $G$ be a connected linear algebraic group, $\beta$ a root of a maximal torus $T$ of $G$, $S = (\textrm{Ker } \beta)^0$, and assume that $Z_G(S)$ is not solvable. Then $T$ is a maximal torus of $...
0
votes
2answers
36 views

How to compute degree of morphism given by a canonical divisor?

Let $X$ be a smooth projective curve of genus $2$. Let $K$ be a canonical divisor. It is known $K$ has degree $2$ and it is not hard to show that $|K|$ is base point free so induces a morphism into $\...
2
votes
1answer
17 views

Why is $s_{\alpha}^{\wedge}(\lambda) = -\lambda$?

Let $G$ be a connected, reductive linear algebraic group with semisimple rank one. Let $H = (G,G)$, $T_1$ a maximal torus of $H$, and $T$ a maximal torus of $G$ containing $T_1$. Let $\lambda: k^{\...
2
votes
1answer
39 views

Extending morphisms between varieties

This is Exercise 5.8 from Gathmann's notes on Algebraic Geometry, and I'm having a bit of trouble for (a) and (b): page 41 of http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/main.pdf (a) ...
3
votes
1answer
73 views

Rational Maps $\mathbb{P}^n \rightarrow \mathbb{P}^m$

In general, any rational map $\mathbb{P}^n \rightarrow \mathbb{P}^m$ can be given by an $m+1$-tuple of polynomials of the same degree, with no common factor. Can anyone give me a reference or a proof ...
1
vote
2answers
105 views

Proof verification affine curve not isomorphic to plane curve

I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve. Here is what I've done: it ...
1
vote
1answer
44 views

Let the ringed space $(X,\mathcal{O}_X)$ be an affine variety. Must $\mathcal{O}_X$ be the structure sheaf?

We say that for a topological space $X$, $\mathcal{O}_X$ is a pre-sheaf if for all open $U\subseteq X$, $\mathcal{O}_X(U)$ is a ring, $\mathcal{O}_X(\varnothing)=0$, and if we have defined $\forall\ U\...
2
votes
0answers
30 views

$h^0(\mathcal O(p)) \leq 1$?

Let $X$ be a smooth projective complex curve of genus $g$ and $D$ an effective divisor. It is true that $l(K-D) \leq g$. I would like to show equality iff $D=0$ or $g=0$. This is Hartshorne IV 1.5. ...
0
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0answers
32 views

Locally free sheaf tensor skyscraper sheaf

Let $X$ be a smooth projective curve over $\mathbb C$ and let $p \in X$. Let $\mathcal F$ be a locally free sheaf of rank 1 over $\mathcal O_X$. Consider the skyscraper sheaf $\mathbb C_p$ at $p$. Is ...
0
votes
0answers
58 views

Why is this sequence exact?

Let $X$ be a smooth complex projective curve. Let $D$ be any divisor and let $p$ be a point. Rick Miranda claims this sequence is exact on page 285 of his book on Riemann surfaces: $$ 0 \to \mathcal ...
1
vote
1answer
45 views

Question regarding proof that a finite morphism is proper.

We know that a morphism $f: X \to Y$ is proper if and only if $Y$ can be covered by open subsets $V_i$ such that $f^{-1}(V_i) \to V_i$ is proper for each $i$. If a morphism is finite we can cover $Y$...
0
votes
1answer
15 views

Is it always true that $N_{(G,G)}(T_1) \subseteq N_G(T)$?

Let $G$ be a connected, reductive linear algebraic group whose semisimple rank is $1$. Then $H := (G,G)$ is a connected semisimple group of rank one. Let $T_1$ be a maximal torus of $H$, and let $T$ ...
3
votes
1answer
166 views

Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
0
votes
0answers
15 views

Closed-form formula for system of two bivariate quadratic polynomials

Given a system of two bivariate quadratic polynomials: \begin{eqnarray} a_0 + a_1 x + a_2 y + a_3 xy+a_4 x^2 + a_5 y^2 &= 0 \\ b_0 + b_1 x + b_2 y + b_3 xy+b_4 x^2 + b_5 y^2 &= 0 \end{...
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0answers
43 views

Smoothness of Schubert Variety

Consider the Schubert variety $X(s_3s_2s_1s_4s_3s_2)$ in $SL_5/P_2$, where $P_2$ is the maximal parabolic corresponding to the simple root $\alpha_2$. In one line notation this permutation can be ...
0
votes
0answers
33 views

how to calculate the implicit Cartesian equation from calabi yau threefold?

i find mathematica version about parametric form, how to calculate the implicit Cartesian equation from calabi yau threefold? ...
3
votes
1answer
56 views

What is the definition of the dimension of an algebraic manifold?

I have a very basic question. It says on Wikipedia that an algebraic manifold is an algebraic variety which is also a manifold. So suppose I have an algebraic manifold $V$ which is an affine variety ...
0
votes
1answer
38 views

Base point free linear system

Let $X$ be a (compact) Riemann surface. Let $D$ be a divisor. In Rick Miranda's book on Riemann surfaces, on page 160, there is a bijection between Base-point-free linear systems of dimension $n$ on ...
0
votes
0answers
30 views

Let $f:X\to Y$ be a continuous map between affine varieties, that pulls back germs of regular functions to regular functions. Then $f$ is a morphism.

More precisely, we have that $\forall\ \alpha\in X, \phi\in \mathcal{O}_{Y,f(\alpha)}$, we have $f^{*}\phi\in\mathcal{O}_{X,\alpha}$, so that $f^{*}\mathcal{O}_{Y,f(\alpha)}\subseteq\mathcal{O}_{X,\...
0
votes
0answers
37 views

Submodules of a free module over a PID - geometry

I've learned that a submodule of a free module $F$ over a PID $R$ is also free. I'm particularly interested in the method given by Martin Brandenburg to solve the question in the following link: ...
0
votes
1answer
35 views

Expressing a hypersurface of a variety as zero locus

It should be obvious from the question that I am not an algebraic geometer, and so I would really appreciate an answer without using schemes or functor. Let $V$ be an (embedded) variety in a complex ...
1
vote
1answer
33 views

Dimension of variety intersected with a hyperplane?

I have an affine variety $V$ defined over $\mathbb{C}$, defined as a zero set of some polynomials in $\mathbb{C}[x_1, ..., x_n]$. Let $$ W = V \cap \{ x_n = 0 \}. $$ I have two questions. Does $W$...
0
votes
0answers
34 views

Let $X\in\mathbb{A}^n$ be a variety, $U\in X$ open, $f\in A(X)$, $f(x)=0\ \forall\ x\in U$, then $f=0$.

I know that an analogous statement holds true in complex analysis, but I don't know how to prove the statement here. I know that $U$ is dense in $X$, and $f:X\to\mathbb{A}^1$ defines a continuous ...
4
votes
1answer
55 views

Is every unramified extension of DVRs simple?

Let $A$ be a discrete valuation ring with maximal ideal $\mathfrak{m}$, fraction field $K$, and $L$ a finite separable extension of $K$ degree $n$, unramified w.r.t. $A$. Let $B$ be the integral ...
3
votes
2answers
71 views

Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
1
vote
2answers
65 views

Shouldn't $t^n : \mathbb{A}^1 \rightarrow \mathbb{A}^1$ ramifies at $0$?

Yo, this is probably the stupidest question ever that I've asked here. Let $$\varphi: \mathbb{A}^1 \rightarrow \mathbb{A}^1$$ be the map of schemes (over a field $k$) such that $\varphi (x) = x^n$. ...
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vote
0answers
39 views

Is there a “strong” Chow lemma where “dense” means “scheme theoretically dense”?

Recall Chow's lemma: Chow's Lemma: If $X$ is a scheme that is proper over a noetherian base $S$ then there exists a projective $S$-scheme $X'$ and a surjective $S$-morphism $f : X'\to X$ that ...
2
votes
0answers
27 views

Tensor product between an invertible sheaf and a constant sheaf.

This question is a natural extension this one. Consider an irreducible scheme $X$ with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf over $X$. Then define the presheaf $$U\...
0
votes
1answer
30 views

“Projective” quotient of $\Bbb{Z}^2$

Consider the space of integer points $\Bbb{Z}^2=\{(x,y)|x,y\in\Bbb{Z}\}$. Consider now the equivalence relation: $$ (x,y) \sim (x',y') \quad \Leftrightarrow \quad \beta x'=\alpha x,\, \beta y'=\...
0
votes
1answer
36 views

Is the plane curve $y^3=x^4+x^3$ an irreducible algebraic affine set?

I'm dealing with the plane curve $C=\{(x,y)\in k^2:y^3=x^4+x^3\}$. I want to know if this curve is irreducible, where $k$ is a commutative field. I know this is equivalent to the ideal $\sqrt{I}$ ...
1
vote
1answer
27 views

Structures in Non-linear Sigma Model

I debated whether this belongs here, or on Physics SE, but all my questions correspond to the algebraic/complex geometry, not physics, so hopefully it's okay here. The non-linear sigma model ...
1
vote
1answer
58 views

De Rham interpretation of $H^1(R,p,\mathbb{C})$

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. This is my question: Is there a way to interpret the relative cohomology group $H^1(R,p,\mathbb{C})$ as a De Rham cohomology group ...
7
votes
2answers
90 views

For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
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0answers
40 views

Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
0
votes
0answers
28 views

Sheafifying direct sum of twists

Let be $X\subseteq \mathbf{P}^r$ a smooth projective variety and let be $\mathscr E$ an invertible sheaf over $X$. Let $$M=\bigoplus_{n\geq 0} H^0(\mathscr E(n))$$ as a module over the polynomial ...
5
votes
1answer
123 views

Map of tangent spaces is the Jacobian in Algebraic Geometry

I need your collected brainpower to help me out. This is going to be long, so grab your favorite beverage and snack. I am working through Görtz and Wedhorn's "Algebraic Geometry I" and I am currently ...