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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1answer
34 views

Equivalent definition of almost geometric quotient

I am trying to prove the following lemma - Lemma - Let $X$ be a variety and let $G$ be an algebraic group acting algebraically on $X$. Let $\pi:X\rightarrow X//G$ be a good categorical quotient. Then ...
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1answer
17 views

Show a homomorphism of local rings of two varieties

Shafarevich, Basic Algebraic Geometry 1, II.1.1 "Prove that the local ring $\mathcal O_x$ of the curve $xy=0$ at $(0,0)$ is isomorphic to the subring $\mathcal O \subset \mathcal O_1 \oplus \mathcal ...
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1answer
54 views

Trying to understand some basic facts about tangent space of Grassmannian.

I am reading Harris's 'Algebraic Geometry: A first course'. I am trying to understand its identification of the tangent space to a Grassmannian. Let $G(k,n)$ be the Grassmannian of $k$-planes in ...
2
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0answers
39 views

Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take ...
3
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0answers
49 views

Short exact sequence on $\mathbb{P}^1$

Let F be a torsion free sheaf of rank $n+4$ over $\mathbb{P}^1$ which fits in the SES $0\longrightarrow\mathcal{O}_\mathbb{P^1}(-3)^{n+2}\longrightarrow ...
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1answer
21 views

Natural map from cokernel of a monad

If I have a monad $$ U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W $$ then there should be a natural map $$ \text{cokernel}(\alpha) \rightarrow W $$ but I can't think of ...
2
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1answer
51 views

Global generation of vector bundles by an exact sequence

Let $X$ be a smooth projective complex surface and $V$ a globally generated vector bundle on $X$. Suppose we have a vector bundle $E$ sitting in an exact sequence $$0\to V\to E\to O_X(C)\otimes A \to ...
0
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1answer
75 views

An inequality about the dimension of fiber

I am working on Problem 11.4.A of Vakil's notes: Let $X$ and $Y$ be two locally noetherian schemes, and $\pi:X \to Y$ is a morphism. $\pi(p)=q$. Then prove: $codim_Xp \leq ...
0
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1answer
51 views

Example of a Zariski sheaf which is not representable?

I am looking for an example of a contravariant functor from $Sch \to Set$ which is a Zariski sheaf, but which is not representable.
3
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1answer
37 views

Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
2
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0answers
61 views

Intersection condition for a Grothendieck topology

I am a bit confused about constructing a Grothendieck topology from a Grothendieck pretopology, largely because I have a discrepancy in definitions of the former. According to all of the questions ...
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2answers
57 views

Reference request on complex projective algebraic geometry

I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic ...
1
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1answer
56 views

A few questions about vector bundles on an algebraic variety

Let $X$ be a smooth projective complex variety and $E$ an algebraic vector bundle on $X$. (Q1) If $E$ is globally generated and $c_1(E)=0$ does it follows that $E$ is trivial? (Q2) If $E$ is ...
1
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1answer
96 views

Vakil FOAG 11.3.B

I am thinking about how to use Krull's PIT to prove this statement (11.3.B on Vakil's notes): If $(A,m,k)$ is a Noetherian local ring with maximal ideal $m$, and $f \in m$, then $\dim A/(f) \geq ...
2
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2answers
59 views

Example of non-noetherian ring whose spectrum is noetherian

Since spectrum of noetherian ring is a noetherian topological space, I am finding an example s.t. a non-noetherian ring whose spectrum is noetherian. Since most nice rings are noetherian, actually I ...
0
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1answer
37 views

Closed points of $Spec(A)$ are dense

This is exercise 3.6.J from the most recent Vakil's notes. Suppose that k is a field, and A is a finitely generated k-algebra. Show that closed points of Spec A are dense, by showing that if f ∈ A, ...
1
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1answer
40 views

Bertini's Theorem and singular divisors on a surface

I'm trying to understand the following: Let $X$ be a projective, smooth surface over an algebraically closed field and $D$ a divisor on $X$. How can I see that $D$ is linear equivalent to the ...
1
vote
1answer
54 views

On the genus of a curve and its set of rational points.

The genus $g$ of a nonsingular curve $C$ of degree $n$ is defined as $g = \frac{1}{2}(n-1)(n-2)$. Let $C(Q)$ denote the set of rational points on $C$. By Faltings, we know that $C (Q) < \infty$ for ...
1
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1answer
37 views

Is the inverse image of an irreducible variety under the natural projection irreducible (in the setting of homogeneous spaces)?

Let $p\colon Z\to X$ be a morphism between irreducible varieties (= reduced schemes of finite type over $\mathbb{C}$). Assume that every fiber of $p$ over a closed point of $X$ is also irreducible. ...
0
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2answers
75 views

Closed subschemes of affine scheme

In Mumford's Red Book (p. 106, 2nd edition, Theorem 3) it is proved that any closed subscheme $(f,f^\sharp):Y→X, f$ the inclusion, of an affine scheme $X=\operatorname{Spec}R$ is of the form ...
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0answers
26 views

Torus action on Schubert varieties

Let $T$ be a torus and $B$ be a Borel subgroup in a simple algebraic group $G$. Then $T$ acts on $G/B$ by left multiplication. Then the action also restrict to any Schubert cell $BwB/B$. But how does ...
1
vote
1answer
36 views

Proof of fibers of lattice are finite

By lattice I mean a subgroup of the additive group $\mathbb{Z}^k$. The lattice $\mathcal{L}$ shall have the property that the only non-negative vector in $\mathcal{L}$ is the origin, i.e. ...
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0answers
43 views

How to make the hyperbola to an abelian variety?

Suppose we are given complex numbers $x,y,\alpha,\beta$ such that $x^2-y^2=1$ and $\alpha^2-\beta^2=1$. Is it possible to find complex numbers $c:=f(x,y,\alpha,\beta), d:=g(x,y,\alpha,\beta)$ such ...
0
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1answer
37 views

Prop. 2.3 Hartshorne: φ:A→B φ A B induces a morphism Spec(B)→Spec(A)

It is on page 73 Prop 2.3. I do not understand a step in the part(c). That is, if we have $X=\operatorname{Spec}(A)$ and $Y=\operatorname{Spec}(B)$, given a morphism of local ringed space ...
2
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1answer
68 views

For a ring $R$, what is $\text{Gr}(R)$?

I'm reading Deligne's "The fundamental group of the projective line minus 3 points", specifically the chapter on tangential base points (15.14), where in 15.20, he suddenly uses the notation $Spec\; ...
2
votes
1answer
35 views

Given $\pi:X\rightarrow Y$ how to show $X$ is irreducible (resp. normal) $\Rightarrow$ $Y$ is irreducible(resp. normal)?

Let $G$ act on the affine variety $X=\operatorname{Spec}(R)$ such that $R^G$ is a finitely generated $\mathbb C$ - algebrs and let $\pi:X\rightarrow Y=\operatorname{Spec}(R^G)$ be the morphism of ...
2
votes
2answers
65 views

Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free ...
0
votes
1answer
25 views

Normalization of ring of polynomials

Let $x_1(t),...,x_n(t)\in\mathbb{C}[t]$ be such that $\mathbb{C}[t]$ is finite as a $\mathbb{C}[x_1(t),...,x_n(t)]$-module and that $\mathbb{C}(x_1(t),...,x_n(t))=\mathbb{C}(t)$. How to show that ...
1
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1answer
70 views

If $L$ is a line bundle on a scheme $X$, what is the ring $\oplus_{n \geq 0} \Gamma(X, L^{ \otimes n})$?

If $L$ is a line bundle on a scheme $X$, what is the ring $A = \oplus \Gamma(X, L^{ \otimes n})$? This ring comes up in an exercise that I am struggling with right now, and I would like some insight ...
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1answer
137 views

What is the geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
2
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0answers
59 views

Invariant ring for $S_5$ [closed]

For the standard representation of $S_5$, the ring of invariants is generated by the elementary symmetric polynomials and hence it is a polynomial ring. Now if we take the tensor product of standard ...
3
votes
1answer
49 views

Why is this a group action - what is the significance of $g^{-1}$?

Let $G$ be a group acting on a variety $X$ such that every $g\in G$ defines a morphism $\phi_g:X\rightarrow X$ given by $\phi_g(x)=g\cdot x$. If $X=\operatorname{Spec}(R)$ is affine then $\phi_g$ ...
0
votes
1answer
33 views

Radical of an ideal in $R [x]$

Let $\frak {I}$ be an ideal of $R[x]$, the polynomial ring over a commutative ring with identity $R$. Is it true that the radical of $\frak{I}$, the intersection of all prime ideals containing ...
0
votes
1answer
40 views

Closed subscheme defined by kernel of diagonal homomorphism

Let $f: X \to Y$ be a morphism of schemes. Let $\Delta : X \to X \times_Y X$ denote the diagonal morphism. Take $U = \textrm{Spec } A$ and $V = \textrm{Spec } B$ be affine open subsets of $Y$ and $X$ ...
6
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1answer
77 views

Is representability of Zariski sheaves local on the base?

Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property: Sheaf condition: For every ...
3
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0answers
37 views

Throwing out embedded points

Suppose $X$ is an irreducible scheme, but maybe generically non-reduced or with embedded points. Is there some natural scheme structure on $X$ that gets rid of the embedded points, but remembers the ...
1
vote
1answer
25 views

Boundary is a union of orbits with strictly lower dimension

I'm stuck on the proof of the following propsition in Humphreys Linear Algebraic Groups: Let $G$ be an algebraic group acting morphically on a variety $X$. Then each orbit is a smooth, locally ...
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0answers
24 views

Show that the pullback $\mathbb{C}[W] \rightarrow \mathbb{C}[V]$ is injective iff $F$ is dominant, that is, the image set $F(V)$ is dense in $W$.

The question is: Show that the pullback $\mathbb{C}[W] \rightarrow \mathbb{C}[V]$ is injective if and only if $F$ is dominant, that is, the image set $F(V)$ is dense in $W$. The $W, V$ are ...
0
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0answers
35 views

Isogeny of elliptic curves over $p$-adic field

If $K$ is a $p$-adic field, and $E_q$ and $E_{q'}$ are the corresponding Tate curves for $|q|,|q'|<1$, why does $E_q$ and $E_{q'}$ being isogenous imply that there are integers $A$ and $B$ such ...
2
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0answers
29 views

Blowing up a double cover at a ramified point

Let $V$ be a projective variety which is a double cover, that is, there exists a finite map $\sigma: V \to Q$ of degree 2, ramified over a divisor $W \subset Q$. Suppose $x \in V$ is smooth (the case ...
0
votes
1answer
28 views

Surjective morphism to Quotient sheaf, nonsurjective on sections

Let $k$ be an algebraically closed field, $X=\mathbb{P}^1$ the projective line. Let $P=(1:0)$ and $Q=(0:1)$ be points on $X$, and $\mathscr{F}$ be the sheaf of regular functions on $X$. Define a ...
2
votes
1answer
67 views

Intuition behind a first Chern class computation

On a complex smooth algebraic surface $X$, say we have a vector bundle $F$ which fits in an exact sequence $$0\to F\to O_X^{r+1} \to A\to 0$$ with $A$ a torsion sheaf supported on a smooth curve ...
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0answers
36 views

Intersection of curves on a surface

I have a question about Lemma 1.3 in Hartshorne (on page 358). He wants to prove that $$\#(C\cap D)=\deg_C(L(D)\otimes \mathcal{O}_C).$$ Here, $L(D)$ is invertible sheaf on X correspond to a curve ...
2
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2answers
67 views

Sum of ideal sheaves commutes with taking global sections

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ effective divisors intersecting each other at finitely many points. Is it true that ...
2
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0answers
28 views

Real points in a matrix interval

Given $A$ and $B$, two $n \times n$ complex Hermitian positive semidefinite matrices such that $A< B$. I want to show existence (or non existence) of a real symmetric positive matrix $X$ such that ...
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2answers
66 views

Two polynomials $f,g \in K[x,y]$ ring. Prove that $K[x,y]/(f,g)$ is finite dimensional vector space

Let $f,g \in K[x,y]$ be polynomials with no common factor. Prove that $K[x,y]/(f,g)$ is a finite dimensional vector space. I know there are non-zero (this word is correct?) $r(x)$ and $s(x)$ in ...
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0answers
38 views

Two questions on dimensions of affine algebraic variety

This is two exercises from An Invitation to Algebraic Geometry: Show that dimension is an invariant of the isomorphism class of a variety. It intuitively makes sense, but the dimension ...
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1answer
56 views

Show that the Zariski topology on $A^2$ is not the product topology on $A^1 \times A^1$. (Hint: consider the diagonal.)

This is an Exercise in An Invitation to Algebraic Geometry by Karen Smith. I'm not sure what the hint means thus have no clue how to approach. Any thought please? Thanks very much!
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1answer
26 views

Removing quintic plane from projective plane.

How to show that $P^2/X$, where $X$ is a plane quintic (i.e a curve of degree 5) is affine. Thanks for any suggestion!
0
votes
1answer
16 views

Linear algebraic group inside $GL_n$

Let $G$ be a linear algebraic group. Consider the closed embedding $G \hookrightarrow GL_n$. Let $K$ be any field. Let $x \in GL_n(K)$. Now suppose we know that $x^n \in G(K) $ for some positive ...