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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2
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1answer
101 views

Is a Variety a manifold?

Is it true that every smooth variety (over $\mathbb{R}$ or $\mathbb{C}$ ) is a (real or complex) manifold? I have tried to show this using the implicit function theorem but I am not getting anywhere. ...
1
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0answers
45 views

Linear Transformation of points [duplicate]

Could you help me solve this: For projective coordinate system on the line $l$ are given points $A (2,1)$, $B (1,1)$, $C (0,1)$, $A_1 (0,1)$, $B_1 (1,5)$ and $C_1 (2,1 )$. Find a linear transformation ...
0
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0answers
32 views

Is it possible to solve this series of triangles with only the given information?

Consider the following: As displayed in the picture, the distance between the points is 1, so the last point's coordinates would naturally be $(c, d+4)$ Is it possible to solve for the coordinates ...
0
votes
2answers
29 views

Solving for points in a plane based on line lengths and geometry

I have the following points and lines in a plane: The problem is this: Given that we know the lengths of lines A, B and C, how can we calculate the coordinates of each point a, b and c? The ...
3
votes
1answer
58 views

Existence of solutions for system of equations

I have a system of equations and was wondering whether there is any obvious reason that you find solutions for $e,f,c$ given a fixed $a \in \mathbb{R}$(which is true). So I don't want to solve this ...
0
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1answer
35 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
0
votes
1answer
70 views

Exercise 5.18(d), chapter 2. Hartshorne

In this exercise, suppose we start with a locally free sheaf $ \mathscr E $ of rank n over a scheme $\mathbf Y $, then corresponding to that we can associate $\mathbf V(\mathscr E^ \lor) $ and when ...
1
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0answers
40 views

Lattice Points On the surface of a sphere

Consider the surface of a $3$-$d$ sphere $S$ with radius $n$ and centered at origin.How can we find all the lattice points on $S$?Thanks in advance!
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0answers
37 views

Why are function fields of $U\subset V$ are equal?

Let $U\subset V$ be two affine varieties. Define the function field of an affine variety $X$ to be $$ k(X)=\operatorname{Frac}(A(X)) $$ How to show that $k(U)=k(V)$? In particular, if ...
1
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1answer
20 views

Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
2
votes
1answer
46 views

Picard group of a smooth projective curve

I have two (related) questions regarding the Picard group: 1) Are there examples of smooth projective curves with large Picard groups (say $Pic(X)\simeq\mathbb{Z}^n)$ for any $n$)? 2) In general, ...
2
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0answers
33 views

Condition on degree to ensure that a generic line bundle of that degree is very ample

Let $C$ be any smooth genus $g\geq 1$ curve (exclude $g=1$ as well if you want). What is the smallest possible integer $d$ to make the following statement correct: "A general line bundle $L$ of degree ...
2
votes
1answer
33 views

Maps between tangent space of $X=\mathbb{V}(x^3-y^2)$ and $Y=\mathbb{V}(x^4-y^3)$

It is an exercise, I am not sure if I am doing right. Let $X=\mathbb{V}(x^3-y^2)$ and $Y=\mathbb{V}(x^4-y^3)$ in $\mathbb{A}^2$. 1) Find the tangent spaces of $X$ and $Y$ at the origin, ...
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votes
0answers
18 views

Non (quasi) affine or projective Variety

Is there a Variety (in the sense of Serre, that is given as a ringed space with affine covering) that is neither embeddable as (quasi) affine, nor (quasi) projective Variety? I believe for Schemes ...
3
votes
1answer
58 views

Subtle aspect of closed subscheme

Let me define a closed subscheme of a scheme $X$ as: An equivalence class of data in the form $$(Z,Y,i,i^\sharp)$$ where $Z$ is a closed subset of $X$, $(i,i^\sharp): Y \to X$ is a morphism of ...
2
votes
3answers
78 views

Algebraic group with no $k$-rational points

Let $k$ be a field, and $G$ an algebraic group. What are some nontrivial examples of an algebraic group with no $k$-rational points?
0
votes
1answer
31 views

The pullback of a nontrivial line bundle is nontrivial?

Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ of rank $2$. Let $\mathbb{P}(E)$ denote the projectivization of $E$, with the natural map $p: \mathbb{P}(E) \rightarrow X$. ...
1
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0answers
14 views

multiplicity computation at a singular point of a surface.

Consider the following quadric hypersurfaces in $\mathbb{P}^5$: $H_1: x_0l_1(x_4,x_5)+q_1(x_1,x_2,x_3,x_4,x_5)=0$, $H_2: x_0x_5+x_1l_2(x_3,x_4,x_5)+q_2(x_2,x_3,x_4,x_5)=0$, $H_3: ...
4
votes
1answer
96 views

Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book ...
2
votes
1answer
47 views

Orientability of algebraic manifolds

Is algebraic manifold always orientable? For example, unorientable Mobius strip $M$ can be represented as $$x(u,v)= \left(1+\frac{v}{2} \cos \frac{u}{2}\right)\cos u$$ $$y(u,v)= \left(1+\frac{v}{2} ...
2
votes
1answer
63 views

Holomorphic Differentials on a non-singular curve.

So I've been working on this for an exam I have coming up and I'm not sure I really understand. If I have a curve defined by some homogenous polynomial P, I can show that the canonical divisor class ...
2
votes
0answers
62 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
6
votes
1answer
49 views

Degree 3 algebraic curve with a triple point

The following problem appeared on an exam I had yesterday. I was unable to solve it, but I would like to know the solution. Let $k$ be algebraically closed, and let $f\in k[X,Y]$ be a polynomial ...
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0answers
20 views

irreducibility of a regular scheme over a local field

Are there any criterions allowing us to detect whether a regular scheme over a local field is irreducible ?
1
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1answer
50 views

Points of a fiber product of schemes

There is a lemma in Stacks Project saying that points $z$ of $X \times_S Y$ are in bijective correspondence to quadruples $$ (x, y, s, \mathfrak p) $$ where $x \in X$, $y \in Y$, $s \in S$ are points ...
5
votes
3answers
71 views

Elements of Spec$(\mathbb{C}[x_1,\dotsc,x_n]/(f_1,\dotsc,f_r))$.

I was reading in Vakil's Foundations of Algebraic Geometry that one can picture the "traditional points" of Spec($\mathbb{C}[x_1,\dotsc,x_n]/(f_1,\dotsc,f_r))$ as the zero locus of the polynomials ...
0
votes
1answer
62 views

Krull dimension of $k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. [closed]

I need help to solve this exercise. If anyone can help, thanks in advance! Let $k$ a field and $R=k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Find the Krull dimension of $R$.
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0answers
39 views

sum of line bundles

Let $f:X\times_k K\rightarrow X$ be the projection for a finite field extension and $X$ lets say a projective (or proper) scheme over $k$. Is it right that one has for a line bundle $L$ on $X\times_k ...
1
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0answers
35 views

The cardinality of the preimage of a point under a nonzero isogeny equals the separable degree of the isogeny

Let $f:E_1\rightarrow E_2$ be a nonzero isogeny between elliptic curves. Take a point $Q \in E_2$. I am looking for a reference to a proof, or a proof, of the following fact: ...
2
votes
2answers
88 views

Two nonassociated functions defining the same hypersurface?

Let $X\subseteq\mathbb P^n$ be a complex, irreducible projective variety. Let $R$ be the projective coordinate ring of $X$, i.e. $R=\mathbb C[x_0,\ldots,x_n]/I$ for some homogeneous prime ideal $I$. ...
4
votes
1answer
74 views

Is the global section ring of a Noetherian Scheme Noetherian as well?

As the title suggests, I am asked to prove that, given a Noetherian scheme $(X,\ \mathcal{O}_{X})$ and any open subset $U\subseteq X$, $\Gamma(U,\ \mathcal{O}_{X}):=\mathcal{O}_{X}(U)$ is a Noetherian ...
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0answers
25 views

$\mathcal{O}_{X}/\ker(\sigma) \cong \mathcal{O}_{X}(Z(\sigma))$, where $Z(\sigma)$ is the zero locus of $\sigma$?

I have a morphism $f: \mathscr{F} \rightarrow \mathscr{G}$ of invertible sheaves in a smooth variety $X$, or equivalently, a global section $s \in H^{0}(X, \mathscr{F}^{\vee} \otimes \mathscr{G})$, ...
1
vote
1answer
41 views

Loci of cubics and intersection theory

Could you help me to understand what does the calculation in the bottom of the image mean? From where does $\tau^*(\zeta)$ appear? It is page 48 from preprint 3264 & All That Intersection Theory ...
0
votes
1answer
48 views

Is the set $\{y\in Y\mid f^{-1}(y)\subset H\}$ closed? [closed]

Suppose $f\colon X\to Y$ is a surjective morphism of projective varieties, $H\subseteq X$ is a closed subset in $X$, does this hold: $\{y\in Y \mid f^{-1}(y)\subset H\}$ is a closed subset of $Y$?
6
votes
0answers
104 views

Some questions related to Iwasawa invariants of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$. Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
0
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0answers
13 views

Applying moment of inertia of a flat plate algorithm as an integration transform/new integration technique?

Currently in my engineering statics course we are learning about moments of inertia of flat plates and something popped into my head. If we have a flat plate in the xy-axis and we can determine the ...
3
votes
1answer
82 views

Problem I.3.13 in Hartshorne - Follow Up

Consider the context of this question: Local Ring of a Subvariety (problem 1.3.13 in Hartshorne). In trying to prove that $\dim \mathcal{O}_{Y,X} = \dim X + \dim Y$, I proved that $\mathcal{O}_{P,X} ...
4
votes
0answers
25 views

Fundamental group of family of rational varieties

Let $X$ be a smooth, projective, simply connected variety over a field $k$ (i.e. $\pi_1^{\text{et}} = 1$). Let $f: Y \to X$ be a family of rational varieties parametrized by $X$, such that $Y$ is ...
5
votes
1answer
57 views

Is a morphism with finite fibers birational?

Let $f: X \rightarrow Y$ be a morphism of projective varieties such that its fibers have finitely many points. Is $f$ birational on its image? Thanks.
4
votes
0answers
39 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
5
votes
1answer
40 views

quasicompact scheme are finite union of affine scheme

This is a problem from Liu`s book. Show that a scheme $X$ is quasi-compact if and only if it is a finite union of affine schemes. If the scheme is quasicompact then it is obviously a finite union of ...
3
votes
0answers
54 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
2
votes
1answer
43 views

Algebraic Compact manifold originates from a proper scheme?

If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over ...
10
votes
2answers
146 views

What is a “subscheme”?

Every source I've looked at defines open subschemes and closed subschemes, but the definitions always look ad-hoc and not closely related to one another. Are there other kinds of subschemes? If not, ...
1
vote
1answer
87 views

How is the induced morphism $(Z,\mathcal{O}_Z)\to (X,\mathcal{O}_X)$ defined?

I'm reading Algebraic Geometry I by Görtz and Wedhorn and have a question concerning the proof of Lemma 1.55. $X$ is an irreducible affine algebraic set and $Z\subseteq X$ is an irreducible closed ...
0
votes
1answer
40 views

Why are elements of $\mathcal{O}_X(U)$ continuous?

I'm reading Algebraic Geometry I by Görtz and Wedhorn. Definition 1.46 is about prevarieties: A prevariety is a connected space with functions $(X,\mathcal{O}_X)$ with the property that there exists ...
-1
votes
1answer
49 views

Open set in the spectrum of a ring?

Consider $Spec(K[X])$ where $K$ is an algebraically closed field. Is $0$ open in the Zariski topology on spectrum? Does the spectrum have points which are neither open nor closed?
1
vote
1answer
49 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
3
votes
0answers
85 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
1
vote
1answer
25 views

regular functions on intersection of two open sets

Consider the projective space $\mathbb{P}^n$ and let $U_i$ be the open set $x_i \neq 0$. Then $U_i \cong \mathbb{A}^n$ under the isomorphism $(a_0,\dots, a_i,\dots,a_n) \mapsto (a_0,\dots, ...