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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4
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1answer
360 views

Example of a restriction of sheaf not being a sheaf

Let $\mathscr{F}$ be a sheaf on $X$, and $Y\subset X$ a subset. Define a presheaf $\mathscr{F}|_Y$ on $Y$ via the direct limit $$\mathscr{F}|_Y(V):=\lim_{V\subset U}\mathscr{F}(U),$$ where $V$ is an ...
3
votes
1answer
81 views

$\mathbb A^n$ is not a complete quasiprojective variety

Suppose that $\mathbb K$ is an algebrically closed field. In the course of Algebraic Geometry I've attended we gave the following definition (in what follows every space is to be intended on the field ...
3
votes
1answer
106 views

Inverse image presheaf

Let $f:X\rightarrow Y$ be a continuous map of topological spaces, and $\mathscr{G}$ a sheaf on $Y$. So far I failed to come up with a simple example where the presheaf $f^{-1}\mathscr{G}$ on $X$ ...
5
votes
1answer
66 views

Algebraic Groups Problem [duplicate]

I am currently trying to go through Humphrey's Linear Algebraic Groups and am stuck on a problem that sounds deceivingly simple but can not seem to figure out (Problem 5 from Section 7). I have spent ...
2
votes
2answers
185 views

Quadratic Bezier curves representation as implicit quadratic equation

A quadratic bezier curve from points P1=(x1, y1) to P3=(x3, y3) with control point ...
2
votes
2answers
37 views

How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$?

Let $V=K^{n+1}$ be a vector space of dimension $n+1$ and $\mathbb{P}V$ the projective space associated to $V$. How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$? ...
8
votes
0answers
145 views

Canonical sheaf not globally generated for a certain surface.

Me and a friend tried the following problem, but with no luck. Anything would be appreciated: Let $X \rightarrow S$ be an arithmetic surface such that for some $s \in S$, $X_s$ is the union of two ...
4
votes
1answer
151 views

What is the stalk at a point of the quotient of a scheme?

This came up when I was talking to @Benjalim on the chat. Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points ...
7
votes
1answer
191 views

Zariski topology of $k^2$

I've found this "proof left to the reader" in some lecture notes: Let $k$ be an arbitrary field. In the Zariski-Topology of $k^2$ every closed set of $k^2$ is either finite or the zero set of a ...
2
votes
0answers
96 views

Galois actions on extensions of algebraic function fields

Let $k$ be an algebraically closed field, $C/k$ and $C'/k$ be smooth projective curves, and $C'/k \rightarrow C/k$ be a $k$-morphism which is corresponding to the field extension $k(C) \hookrightarrow ...
5
votes
2answers
200 views

Sheafication of a sheaf restricted to a open set

Let $X$ be a topological space and $U$ be open in $X$. Let $\mathcal F$ be a presheaf of rings on $X$. Let $\mathcal F_u$ denote the presheaf restricted to the open set $U$. $\mathcal F^+$ denote the ...
6
votes
1answer
312 views

Does an invertible sheaf always have global generating sections?

Theorem 7.1 in chapter 2 of Hartshorne's text says that invertible sheaves on a scheme $X$ together with its given global generating sections correspond to morphisms from $X$ to $\mathbb{P}_A^n$ (here ...
4
votes
1answer
140 views

example of locally finite type not finite type

Let $A$ be a ring then, a homomorphism $A\rightarrow A[x_1,\cdots,x_n]$ induces a finite type morphism between spectrums. I want to find the map which is a locally finite type but not finite type.... ...
1
vote
1answer
65 views

Existence of prequantization on a simply connected manifold

Let $M$ be a simply connected manifold. Then when, $M$ has a unique pre-quantization and when there is no pre-quantization on $M$.
0
votes
1answer
92 views

pre-quantization on cotangent bundle $T^*M$

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?
0
votes
1answer
51 views

Algebraic Arc Definition

I am searching for an elegant/compact definition/description of an arc into the algebraic domain given 2 points and ...(?) that allow to define every arc. I know it is possible to define the arc given ...
0
votes
1answer
39 views

If the * of morphisms (poly. maps) are equal, are the morphisms equal?

Let $t,s:X\rightarrow Y$ be polynomial maps between affine varieties and $t_*,s_*:k[Y]\rightarrow k[x]$ be their images under the representable contravariant functor. We've learnt that for any ...
2
votes
1answer
172 views

Stalk of the sheaf of regular functions on a subvariety

Suppose $Y$ is a subvariety of a variety $X$ (according to Hartshorne this means if $X$ is quasi-affine or quasi projective then $Y$ is a locally closed subset of $X$, c.f. exercise 3.10, chapter 1). ...
8
votes
0answers
247 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
3
votes
1answer
348 views

Some questions on the basics of invertible sheaves

Let $X$ be a scheme. A $\mathcal O_X$-module $\mathcal L$ is called invertible if, for every point $x\in X$, there is an open neighborhood $U$ of $x$ and an isomorphism of $U$-modules $\mathcal O_X|_U ...
6
votes
2answers
112 views

When a value of a polynomial over $\mathbb Z$ is a perfect square

For which values of $x\in\mathbb{Z}$ the polynomial $16x^3-24x+9$ is a perfect square? I don't know if this question has a solution, but Wolfram Alpha says that the answer is $x=0$ (click), even if ...
4
votes
1answer
273 views

Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
3
votes
1answer
173 views

what does project away mean?

I realise I should know this but I have no idea what people mean when they say "we project away from this point" (or replace point with line, plane or whatever in projective space). What does this ...
2
votes
0answers
63 views

an argument regarding the dimension of a real algebra

Let $S$ be an algebraic subset of $\mathbb{R}^d$ with vanishing ideal $I_S=(f_1(x),\cdots,f_m(x))$. Suppose that $S$ has infinite cardinality (countable or uncountable, does not matter). I want to ...
6
votes
1answer
313 views

Trying to understand open (closed) subfunctors

I am trying to read about functor of points and I am struggling with the definition of open subfunctor. The definition is the following. A subfunctor $\alpha\colon G\to F$, where $F,G\in ...
2
votes
2answers
158 views

Cover of a Grassmannian by an open set

I am reading this document here and in exercise 1, the author asks to show the Grassmannian $G(r,d)$ in a $d$ dimensional vector space $V$ has dimension $r(d-r)$ as follows. For each $W \in G(r,d)$ ...
9
votes
4answers
460 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 ...
4
votes
2answers
546 views

Definition of formal neighbourhood

Consider the scheme $\mathbb{P}^1$, and the point $0 \in \mathbb{P}^1$. What is the formal neighbourhood of $0$ in $\mathbb{P}^1$? Or if you know a good reference, that would be helpful.
0
votes
1answer
62 views

How do I show that the coordinate ring $K[V]$ is isomorphic to $K[p_1\mid V,\ldots,p_n\mid V]$

Let $K$ be an algebraically closed field. Let $I:PK^n \rightarrow PK[x]$ be the standard map from subsets to ideals. ($P$ here is the power-set functor - so that is $PK^n$ is all subsets of $K^n$ and ...
4
votes
1answer
252 views

Algebraic Solutions to Systems of Polynomial Equations

Given a system of rational polynomials in some number of variables with at least one real solution, I want to prove that there exists a solution that is a tuple of algebraic numbers. I feel like this ...
2
votes
1answer
147 views

$f(Y)$ is closed iff stable under specialization

Let $f\colon Y \to X$ be a quasicompact morphism of schemes and suppose that $f(Y)$ is stable under specialization. Then $f(Y)$ is closed. I'm trying to follow the proof given here ...
5
votes
0answers
186 views

generic regularity of affine varieties

Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points? In view of the Jacobian ...
12
votes
1answer
520 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
5
votes
1answer
701 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 ...
7
votes
1answer
134 views

Isomorphic varieties

I just want to see if my approach for this problem is fine: Show $W=\mathbb{P}^1 \times \mathbb{P}^1$ is not isomorphic to $W'=\mathbb{P}^2.$ Well $V= \{ [0:1] \} \times \mathbb{P}^1, V' = \{ ...
2
votes
2answers
154 views

A canonical homomorphism of sheaves of modules

Let $\mathcal F$ be a sheaf of $\mathcal O_X$ modules on a scheme $X$. Fix an affine open subset $U$. If $M$ is a module over the coordinate ring of $U$, we let $\tilde{M}$ denote the associated sheaf ...
9
votes
0answers
188 views

Understanding Bertini's theorem

Let's suppose that I am given a pencil generated by the vector fields $X$ and $Y$ in $\mathbb{C}^2$, $\{ Z_\lambda \}_{\lambda\in\mathbb{P}^1}$, that is, $$ Z_\lambda = X + \lambda Y $$ Assume that ...
2
votes
0answers
68 views

If the intersection of the preimage of a generic point of a flat morphism with an open set is dense, does it imply the open set is dense?

Let $f: E \rightarrow M $ be a flat morphism of varieties (over $\mathbb{C}$) and $E^{\prime}$ an open subset of $E$. Assume that both $E$ and $E^{\prime}$ have pure dimension $2n$, where $n$ ...
1
vote
1answer
183 views

What is the definition of a flat morphism?

When we say that a morphism $f: E \rightarrow M $ between two algebraic varieties (over $\mathbb{C}$) is a flat morphism, what does it mean? Does it mean that that the "dimension" of every fiber ...
2
votes
1answer
81 views

Hilbert space on line bundle

Suppose that $L$ is a complex line bundle on a manifold $M$ with measure $\mu$, How can we prove, $L^2(M,L,\mu)$ is Hilbert space?
4
votes
0answers
118 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
4
votes
0answers
101 views

serre duality and logarithmic differentials

Let $D$ be a normal crossings divisor on some smooth projective variety $X$ (say over the complex numbers) and let $\Omega^p_X(\log D)$ be the sheaf of logarithmic $p$-forms. It is $$ \Lambda^p ...
8
votes
2answers
93 views

Is epimorphism preserved under taking sections?

Let $\phi:\mathscr{F}\rightarrow\mathscr{G}$ be en epimorphism of sheaves (say, of rings) on $X$. Is it true that $\phi(U):\mathscr{F}(U)\rightarrow\mathscr{G}(U)$ is an epimorphism of rings for every ...
6
votes
1answer
142 views

transform into weierstrass-form

How can I transform the elliptic curve $E/\mathbb{C}$ of the form $$y^2=4(x-e_1)(x-e_2)(x-e_3)$$ with $e_1>e_2>e_3\in\mathbb{R}$ roots of $E$ into a Weierstrass-Form $y^2=x^3+ax+b$?
2
votes
1answer
249 views

Bertini's theorem: reduction to pencils

I'm studying the proof of Bertini's theorem on "Principles of Algebraic Geometry" by Griffiths and Harris (page 137). The statement is as follows: The generic element of a linear system is smooth ...
4
votes
1answer
63 views

Dimension of range of an function

Let $f$ be a rational function from affine variety $X$ to affine variety $Y$. Is it always true that $\dim X \geq \dim f(X)$? If it is can someone provide me with a proof of it? To me, this is ...
4
votes
1answer
106 views

Question regarding Hartshorne Example II.(6.5.2)

Let $k$ be a field, let $A=k[x,y,z]/\langle xy-z^2\rangle$ and let $X=\operatorname{Spec}A$. Let $Y:y=z=0$ I want to know the divisor of $y$ In Hartshorne book, because $y=0 \Rightarrow z^2=0$ and ...
4
votes
2answers
91 views

Problem 3.1.2 in Liu — Omission in problem statement?

Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows. Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$, let $f(T):X(T)\rightarrow Y(T)$ denote the ...
2
votes
1answer
43 views

Extend maps between etale groups

Let $V$ be a discrete valuation ring, $S=\operatorname{Spec}(V)$ and $\eta$ (resp. $s$) be the generic (resp. closed) point of $S$. Let $G$ and $H$ be flat group schemes over $S$ and assume I know ...
8
votes
1answer
104 views

Curve over ring, covered by two affines

The reference is http://www.math.columbia.edu/~masdeu/files/notes/FallSeminar.pdf, page 9: Let now $C/R$ be a curve over a noetherian ring $R$; this means that $C$ is smooth, connected, integral, ...