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Questions tagged [algebraic-geometry]

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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13 views

Preimage of a prime ideal in a ring of formal power series.

Let $k = \overline{k}$ be a field. We have the inclusion $\iota:k[x,y] \to k[\![x,y]\!]$ and the prime ideal $\mathfrak{p} = (y - \sum_{n\geq 1} x^n/n!)$. This ideal is prime, because it is the kernel ...
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0answers
41 views

Prove $\mathbb{C}[x,y]^{G} \cong \mathbb{C}[u,v,w]/(uv−w^{n})$

Define $\mathbb{C}[x,y]^{G}=\{ f \in \mathbb{C}[x,y] ; g.f=f \forall g \in G\}$ where $G=\mathbb{Z}/n\mathbb{Z}, g$ is a generator in $G$ and $G$ acts on $\mathbb{C}^{2}$ by $g.(x,y)=(\omega x, \omega^...
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0answers
39 views

Proving a smooth plane curve?

Can someone provide a proof that the vanishing set of an irreducible, degree 2, homogeneous curve in $\mathbb{C}[x,y,z]$ is a plane curve in projective 2-space and moreover that the plane curve is ...
2
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1answer
40 views

Relative homology and limit

Let $X$ be a smooth manifold and $O\subset X$ be an closed set containing a non-trivial neighbourhood of $x\in X$. The reason to ask the question is to clarify the relationship between limit and ...
1
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2answers
56 views

Prove a set $V$ is not algebraic

I need help showing that the set $V = \{(a,b) \in \mathbb{A}^2(\mathbb{C})\ \vert \ \vert a\vert^2 + \vert b\vert^2 = 1\}$ is not an algebraic subset of complex affine 2-space. I believe that I can ...
1
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1answer
27 views

Rational function representation in the field of fractions of an integral domain

In the book Algebraic Groups and Differential Galois Theory by Crespo and Hajto, one considers $V := V(Y^{2}-X^{3}+X) \subset \mathbb{A}_{\mathbb{C}}^{2}$ and $P(0,0) \in V$. The example claims that $...
3
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2answers
45 views

Finding elliptic curves achieving the upper and lower bounds of Hasse's Interval

I always thought that Hasse's bound is sharp (at least for elliptic curves). In other words I always thought that given a prime number $p$, I can find two elliptic curves $E_1,E_2$ over $\mathbb F_p$ ...
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0answers
20 views

Condition for $(1,1)$-classes on complex tori

Let $X = V/\Lambda$ be a complex torus, where $V$ is a complex vector space and $\Lambda \subset V$ is a full-rank lattice. We can identify the cohomology group $H^{2}(X, \mathbb{Z})$ with ...
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0answers
22 views

Proof check and clarification.

$S$ is a locally Noetherian scheme and $s \in S$ . $Y$ is a scheme of finite type over $Spec \mathcal O_ {S,s} $ I have to show there exists open $U$ containing $s$ and a scheme of finite type $X \...
4
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0answers
23 views

Linear polynomial whose multiplication with a given quadratic polynomial vanishes in the Jacobi ring

Let $\mathbb C[x_1,\ldots,x_n]$ be the polynomial ring of $n$ varibles, and $\mathbb C[x_1,\ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $F\in \mathbb C[x_1,\ldots,x_n]_3$ be a homogeneous ...
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0answers
41 views

Examples of Riemann-Roch application

The RR Theorem is often quoted as a means to constrain rational functions on curves. I would like to see easy examples of application of this theorem for this purpose on the Riemann Sphere $\mathbb{P}^...
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0answers
73 views

Proposition 1.1 of A Royal Road to Algebraic Geometry (Holme)

I've been reading Holme's A Royal Road to Algebraic Geomtery, but I can't seem to prove the first proposition. It is stated (on page 8) as: Proposition 1.1 Given $n+2$ points $P_1,P_2,\ldots,P_{n+2}\...
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0answers
43 views

Not understanding a proof about coherent sheaves on projective schemes in Hartshorne

I have been stuck on the proof of the following statement for a while now. Let $S$ be a graded noetherian domain which is finitely generated by $S_{1}$ as an $A$-algebra where $A = S_{0}$ is a ...
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0answers
31 views

Is Hom scheme between projective curves of large genus finite etale?

Let $K$ be a number field, $T$ be a finite set containing some finite places of $K$, and $S=\operatorname{Spec} O_{K,T}$. If $X,Y$ are two projective smooth curves over $S$ with genus large than $1$ ...
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0answers
27 views

Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
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0answers
32 views

Existence of mod $m^n$-points for every $n$

Let A be a complete noetherian local ring and $m$ be its maximal ideal. If I have some polynomials $f_i$ with coeffecients in $A$, and they have a common zero $x_n$ in $A/m^n$ for every $n$, then must ...
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31 views

The solution if possible [on hold]

Let F=F3[x](x^2+2x-1) where F3 is the field with 3 element ,which of the following statment are true? F is field with 27 elements?
2
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1answer
38 views

Quasicoherent sheaves on the groupoid of vector bundles on a surface

Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$\mathcal L\in QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a ...
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1answer
42 views

Automorphisms of the projective space.

This is a follow-up question to this one: Rational functions on curve In that setting, assume $X=\mathbb{P}^1$ and $f$ an isomorphism, so that we are looking at automorphisms of $\mathbb{P}^1$. ...
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0answers
31 views

Finitely many zeros and poles for a function in a function field of a smooth curve

Let $\bar{K}$ be a perfect field and let $f \in \bar{K}(C)$ be a nonzero function in the function field of $C$, a smooth curve (projective variety of dimension 1). I'm trying to understand why (even ...
2
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1answer
38 views

“line at infinity” in projective plane

("Algebraic Geometry: A Problem Solving Approach" by Thomas Garrity) I am struggling with Exercise 1.4.12.1 in the above, which I quote with some context: Here is my intuitive thinking: (a) lines ...
1
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1answer
43 views

Does the tangent bundle of the projective space split?

Does the tangent bundle of the projective space $\mathbb P^n$ over an algebraic closed field $k$ split i.e can be written as direct sum of two vector bundle of positive rank? For cotangent bundle, ...
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1answer
41 views

Determining ideals, isomorphic rings of $\Bbb C[x, y]/(y^2 - x^3)$?

I've been having a substantial amount of trouble trying to understand the workings of $\Bbb C[x, y]$ mod... anything really. I figure this particular example is a good one to ask here because I ...
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0answers
19 views

Algebraic Groups Connected and Reduced?

Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki https://en.wikipedia.org/wiki/Group_scheme Let me quote it: "... Group schemes ...
3
votes
2answers
66 views

Order of $\operatorname{Gal}(K_s/K_\ell)$

I am reading the proof of Grothendieck’s proposition about $\ell$-adic representations of the decomposition group of some discretely valued field, the proposition in the appendix of Serre and Tate’s ...
1
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0answers
32 views

When does the link of an algebraic singularity determines it algebraic type?

Let $X \subset \Bbb C^n$ be an algebraic hypersurface with an isolated singularity $x$. I know that when $X$ is a curve, $\mathcal O_{X,x}$ is determined by the link of $X$ at $x$ (considered as a ...
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0answers
26 views

Why is the radical ideal of the homogenization of an ideal the homogenization of the radical of said ideal?

Let $I$ be an ideal. If $I^{h}$ is the homogenization of that ideal, defined by $I^{h}:= \langle f^h: f \in I\rangle$ that is the ideal generated by the homogenized elements of I, then why is $\sqrt{I^...
6
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0answers
63 views

Continuity of the Euler characteristic with respect to the Hausdorff metric

Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means ...
2
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0answers
48 views

Smooth curves of odd genus

Let $C$ be a smooth curve of genus $g$ and $J_C$ its intermediate Jacobian. Recall that $J_C$ is a ppav of dimension $g$. Fixing a point $p\in C$, one can define the Abel-Jacobi map $$a\colon C\...
3
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0answers
36 views

Graded global sections of Proj(S) for S a polynomial ring and more general

Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}...
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0answers
43 views

Can every connected reductive group over $\mathbb F_p^{alg}$ be defined over $\mathbb F_p$?

If I have a connected reductive group $G$ over a field with characteristic $p>0$ (for instance the algebraic closure of $\mathbb F_p$), can it always be defined over $\mathbb F_p$? For groups like $...
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1answer
29 views

Morphism from a scheme to the spectra of global section

This is a small inquiry, but I just want to be sure about it. Let $X$ be a scheme. I am trying to understand what it is meant by the cannonical morphism between $X \rightarrow Spec (\Gamma(X,\mathcal{...
2
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0answers
42 views

Use of the Rellich Lemma in the proof of the Hodge Theorem

I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100. The method of proof is to establish a weak solution of ...
1
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1answer
64 views

Reference for algebro-geometric definition of the $\cup$-product in (sheaf) cohomology.

Can anyone give me a reference on how the $\cup$-product of sheaf cohomology is defined? I read somewhere that it has to do with the Yoneda pairing of Ext, but my naive approach did not work, because $...
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0answers
30 views

Is affine variety $U= \mathbb{A}^2_{k} \setminus {0}$? [duplicate]

I'm trying to prove, that $U=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Via $r \colon \mathcal{O}_{\mathbb{A}^2_{k}}(\mathbb{A}^2_{k}) \to \mathcal{O}_{\...
3
votes
1answer
89 views

Rational functions on curve

Consider the following setting: $X$ an irreducible smooth proj. curve and $f$ a morphism $X \rightarrow \mathbb{P}^1(k)$, with $k$ algebraically closed. Call $C$ the complement of $f^{-1}\{(1:0)\}$ ...
1
vote
1answer
29 views

Homogeneous prime ideals and grade zero elements

Let $R = \bigoplus_{d \in \mathbb{N}_0}R^{(d)} $ be a graded ring and for homogeneous ideals $I$, let $V_{proj\;R}(I) = \{p \supseteq I \mid p \text{ is a homogeneous prime ideal and } p \nsupseteq R_+...
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0answers
48 views

Viewing algebraic varieties as Homomorphism

Let $k$ be an algebraically closed field, And let $V(\mathfrak{a})$ be the algebraic variety generated by the ideal $\mathfrak{a} \subset k[x_1,\cdots,x_n]$. I have read that we can identify $V(\...
1
vote
1answer
20 views

When is it the case that all closed immersions of all irreducible components are flat?

Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,\ldots,X_r$ be the irreducible components of $X$ and let $f_i: ...
0
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0answers
41 views

Local properties concerning finitely generated modules over a noetherian ring

Given a commutative unital noetherian ring R and p a prime ideal of R, if M,N are finitely generated R-modules and Mp,Np are isomorphic, then Mf≃Nf for some f∈A−p. How to prove this?
2
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1answer
59 views

Base Change of Algebraic Group

I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of finite type over a field"(p. 17). Here the excerpt: Let $G$ be a ...
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1answer
23 views

Prove that any divisor of order 0 on non-singular projective curve of genus $g$ is equivalent to other

Could you please check whether the solution below is ok? There is an exercise from Shafarevich's Basic Algebraic Geometry, vol. 1, ex. 7.7.21. Let $o$ be a point of an smooth algebraic curve $X$ of ...
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0answers
59 views

Irreducible projective varieties and morphisms

Let $C$ be a smooth and irreducible projective curve, and let $f: X \rightarrow \mathbb{P}^1(\mathbb{C})$ a morphism of varieties: then $f$ is either constant or surjective. I am trying to prove this ...
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0answers
33 views

Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
8
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0answers
56 views

An abelian variety not isogenous to a Jacobian

In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $\mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a ...
2
votes
1answer
62 views

Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.

Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$. Goal: Prove that $p$ is irreducible. Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by $$I:=(p).$$ My approach is to show that ...
1
vote
1answer
34 views

Filtration in crystalline Poincaré Lemma

I am trying to understand section 20 in https://stacks.math.columbia.edu/download/crystalline.pdf, especially the proof of Lemma 20.2. If $A\rightarrow B$ is a map of rings and $P=B[x_i]$ is some ...
2
votes
1answer
36 views

Adjoint property of $f^{-1}$ and $f_*$.

Let $f: X \rightarrow Y$ be a continuous map. Then a standard exercise is to show that the functors $f_*: \text{Sh}(X) \rightarrow \text{Sh}(Y)$ and $f^{-1}: \text{Sh}(Y) \rightarrow \text{Sh}(X)$ are ...
2
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0answers
35 views

Infinitely many exceptional divisors

Does there exist a smooth, complex, projective 3-fold, with infinitely many divisors isomorphic to $\mathbb{CP}^{2}$ which have normal bundle $\mathcal{O}(-1)$?
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64 views

Can the complex numbers be extended from an unsolvable equation in the complex domain? [closed]

In reference to the recently published pre-print https://arxiv.org/abs/1811.12175. The author proposes the extension of the complex numbers from an unsolvable equation in the complex domain, for ...