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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3
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1answer
27 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in ...
0
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2answers
85 views

Cohen-Macaulay and regular rings

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
0
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1answer
28 views

$\dim N_1 X$ and $\mathbb{P}^2$ and $\mathbb{F}_n$

If $X$ is a smooth projective rational surface such that $-K_X$ is big, then why is it that if $\dim N_1X \leq 2$, then $X \cong \mathbb{P}^2$ or $X \cong \mathbb{F}_n$, a Hirzebruch surface. I ...
1
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0answers
24 views

line bundle descents?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
1
vote
3answers
37 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that ...
1
vote
1answer
464 views

Why does surjectivity of the induced map show that a morphism of affine varieties has closed image?

Let $\phi : X \rightarrow Y$ be a morphism of affine varieties and let $\phi^\ast : k[Y] \rightarrow k[X]$ be the induced map on coordinate rings. My text says that if $\phi^\ast$ is surjective then ...
2
votes
1answer
69 views

Pascal's theorem by Bezout's theorem

I need to prove the following theorem Let the hexagon $ABCDEF$ be inscribed in the nondegenerate conic $q=V(f)$. Assume that $A,B,C,D,E,F$ are distinct. Let $P=\overline{FA}\cap \overline{CD}, ...
2
votes
1answer
37 views

What does the spectrum of the Grothendieck ring of varieties look like?

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation ...
2
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0answers
54 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
3
votes
1answer
41 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
0
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1answer
43 views

Presheaf image of a monomorphism of sheaves is a sheaf

EDIT: The original version regarding coker is false. Consider a morphism of sheaves of abelian groups $\Phi:\mathscr{F}\to\mathscr{G}$. It is not in general true that Im$_{pre}\Phi$ is a sheaf. ...
2
votes
1answer
55 views

Colimit preserves monomorphisms under certain conditions

I know that colimit preserves epimorphisms. Consider the special case where The diagrams are indexed by a directed set $I$, We are in the category of certain algebraic structures, such as ...
0
votes
1answer
32 views

How to prove $V(5x^2+6xy+2y^2-2yz-z^2)$ is empty

Let $V/\mathbb{Q}$ be the projective variety $V:5x^2+6xy+2y^2=2yz+z^2$. I want to prove $V(\mathbb{Q})$ is empty. Given $[x,y,z]$ in $V$, WLOG assume $x,y,z\in \mathbb{Z}$ and $\gcd(x,y,z)=1$. ...
1
vote
1answer
32 views

Open in Zariski but not product topology

Let $X \subset k^m , Y \subset k^n$ be algebraic sets ($k$ an algebraically closed field). Then $X\times Y \subset k^{m+n}$ is an algebraic set whose Zariski topology is finer than the product ...
0
votes
1answer
23 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
1
vote
0answers
51 views

Closed sets flat morphism

Consider the sets $Y_{k}:=\{y\in Y\mid |f^{-1}(y)|\leq k\}$, where $f:X\to Y$ is a flat quasi-finite morphism between smooth irreducible affine varieties and $k$ a natural number. Are these sets ...
0
votes
1answer
37 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
3
votes
1answer
185 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
2
votes
1answer
58 views

Inverse image of a line bundle on $\mathbb{P}^1$ and Euler-like exact sequence

Let $E=\mathcal{O}_{\mathbb{P}^1}(-1)$. Then we have the following exact sequence $$0\rightarrow E\rightarrow\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}\rightarrow E^{-1}\rightarrow0.$$ This sequence can ...
0
votes
0answers
29 views

Principal divisor of rational functions over nonsingular curves and pullback

I'm studying the theory of divisor over algebraic varieties for a seminar and I came across a problem that I think I solved almost completely except for a point that I'm missing. Let be $k$ an ...
2
votes
1answer
45 views

First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
3
votes
1answer
67 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
0
votes
0answers
26 views

Non-Singular Abstract Curve

Let $X$ be an affine non-singular curve, over the closed field $k$, with function field $K$. We define $C_K$ to be the collection of all discrete valuation rings of $K$ which contain $k$. For ...
0
votes
1answer
23 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
1
vote
2answers
22 views

Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
1
vote
1answer
106 views

Generalization of Bézout theorem in higher dimension.

I am wondering if there is a generalisation of Bézout theorem in higher dimension. By Bézout theorem I mean: If $f_1,f_2\in\mathbb{C}[X_1,X_2]$ such that $\dim V(f_1,f_2)=0$, then $\#V(f_1,f_2)\leq ...
3
votes
1answer
90 views

Morphisms from spectra to schemes

Let $X$ be a scheme. Show that for any $x \in X$ there exists a canonical morphism $\textrm{Spec}\, \mathcal{O}_{X,x} \rightarrow X$. If $k(x)=\mathcal{O}_{X,x}/\frak{m}_{x}$ is the residue field at ...
6
votes
2answers
199 views

Scheme over S and morphisms

Quoting from Hartshorne Let $S$ be a fixed scheme. A scheme over $S$ is a scheme $X$, together with a morphism $X \to S$. If $X$ and $Y$ are schemes over $S$, a morphism of $X$ to $Y$ as schemes ...
1
vote
1answer
70 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
3
votes
1answer
64 views

Morphisms of schemes

I know that there is a morphism of schemes between $\mathbb{A}_k^{n+1}-\{0\}$ and $\mathbb{P}_k^n$, where $k$ is a field, given by $$(x_0,...,x_n) \to [x_0,...,x_n].$$ But, how can I stricly prove ...
3
votes
1answer
152 views

Graph morphism for a separated morphism of schemes

I want to prove the following: Let $f: X \rightarrow S$ be a separated morphism of schemes. Show that any section $g: S \rightarrow X$ of $f$ i.e. a morphism such that $f \circ g=\textrm{id}_{S}$ is ...
2
votes
1answer
163 views

Separated Morphisms of Schemes

(a) Let $f:X \rightarrow S$ be a separated morphism of schemes. Show that for any subscheme $U \subset X$, the restriction $f\mid_{U}:U \rightarrow S$ is separated. (b) Let $R$ be a commutative ring ...
2
votes
1answer
586 views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
4
votes
1answer
115 views

“This property is local on” : properties of morphisms of $S$-schemes

I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them ...
3
votes
1answer
38 views

When is the symmetric algebra of a vector bundle finitely-generated?

Let $X$ be a projective variety over a field $k$, and $\mathcal L$ a vector bundle on $X$, i.e. a locally free $\mathcal O_X$-module of finite rank. For each $n\geq 0$, $\text{Sym}^n \mathcal L$ is a ...
7
votes
1answer
139 views
+200

What's the difference between cohomology theories of varieties and topological spaces

There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for ...
3
votes
1answer
50 views

How Appell-Humbert theorem works in the simplest case of an elliptic curve

Line bundles on complex tori $V/\Lambda$ could be described by a pair $(H, \chi)$, where $H$ is a hermitian form on $V$ s.t. $\operatorname{Im} H(\Lambda, \Lambda) \subset \mathbb{Z}$, and $\chi$ is a ...
0
votes
0answers
55 views

Curves that don't have lines as components

In my lecture notes we have the following: A point $P=\left [x, y, z\right ]$ of an algebraic curve $C_F=V(F)$ is called an inflection point of $C_F$ when $P$ is not a singular point of $C_F$. ...
1
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2answers
60 views

Number of elements in fiber

My Question: If we have $f:X\to Y$ an etale morphism and we assume $X,Y$ smooth affine Varieties, why is it true, that $|f^{-1}(y)|\leq deg(f)$ ? Why isn´t there any point of $Y$, which has more ...
2
votes
2answers
46 views

A doubly ruled surface which is not a plane must be quadratic

I want to show that a doubly ruled surface which is not plane must be quadratic. Any help will be appreciated.
4
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0answers
35 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
-2
votes
1answer
34 views

how would i answer this question [on hold]

Please help me in this question. Let $$Y=m² - 4n²$$ $$m= 2x + 3$$ and $$n = x-1$$ Find $y$? I have tried it so many times but is not working. I would be very thankful...!!!
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0answers
21 views

equivalent definitions

If $\pi :C^{'} \rightarrow C$ is a double unramified cover of a complex Riemann surface named $C$, we can define the involution sheet exchange $\tau: C^{'} \rightarrow C$. We say that a meromorphic ...
2
votes
1answer
33 views

The classes are lines of $K^3$ that passes through $(0, 0, 0)$.

In my lecture notes we have the following: We consider $(K^3)^{\star}=K^3 \setminus \{(0, 0, 0)\}$ and we define the relation $$(a_1, b_1 , c_1) \sim (a_2, b_2, c_2) \Leftrightarrow (\exists ...
2
votes
2answers
46 views

Lines in the projective plane

In my lecture notes we have the following: The set $$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$ is called projective plane over $K$. There are the following cases: $z \neq ...
1
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0answers
22 views

The order of the composition

Suppose that we have a meromorphic function $f$ defined on a complex riemann surface $C$ and $g$ a holomorphic function from $C$ to $C$ such that the composition is well defined. Write $ord_{p}(f ...
3
votes
1answer
49 views

Computation of the global sections of a normal sheaf

Let $Y\subset X=\mathbb{P}^r$ be the image of the Veronese embedding $\mathbb{P}^1\rightarrow\mathbb{P}^r$. I want to calculate $dim$ $H^{0}(C,\mathcal{N}_{Y|X})$, where $\mathcal{N}_{Y|X}$ is the ...
0
votes
0answers
31 views

Genus 2 Elliptic curves & their periods

The first part of my question is just a check of my knowledge on elliptic curves. I'm fairly happy with the number theory side of things (torsions, rank, whatever) but is my understanding of the more ...
1
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0answers
35 views

An order relation

Suppose that $\pi:C^{'} \rightarrow C$ is a double unramified cover of a riemann surface of genus $g>0$. Let $\tau:C^{'} \rightarrow C$ the involution sheet exanche and suppose that $f$ is a ...
4
votes
2answers
42 views

affine algebraic subset of $\mathbb{A}_k^4$

How do I go about proving the subset $V = \{(s^3, s^2t, st^2, t^3)\text{ }|\text{ }s, t \in k\}$ is an affine algebraic subset of $\mathbb{A}_k^4$ and find $\mathbb{I}(V) \subset k[x_0, x_1, x_2, ...