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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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. ...
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16 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 ...
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3answers
33 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 ...
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1answer
28 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 ...
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1answer
31 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 $\gcd(x,y,z)=1$. ...
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1answer
40 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. ...
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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 ...
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1answer
21 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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1answer
36 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 $Q=-P$ ...
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0answers
48 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 ...
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1answer
44 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 ...
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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 ...
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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 ...
2
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0answers
52 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 $\Phi$ ...
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2answers
21 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 ...
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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 ...
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1answer
57 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 ...
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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 $C_F$. ...
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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 ...
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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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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 ...
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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 ...
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2answers
59 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 ...
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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 ...
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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 ...
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30 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 ...
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0answers
34 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 ...
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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.
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1answer
54 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 ...
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1answer
37 views

Skyscraper sheaf in a s.e.s

On a curve $C$, if $\mathbb{C}_p$ is the skyscraper sheaf at a point $p \in C$, then we have the exact sequence $0 \to \mathcal{L}(-p) \to \mathcal{O}_C \to \mathbb{C}_p \to 0$. On a variety with ...
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2answers
41 views

Pullback commutes with dual for locally free sheaf of finite rank

Let $ f:X\rightarrow Y$ be a morphism of ringed spaces. Let $ \mathscr{E} $ be an $\mathcal{O}_Y$ module that is locally free of finite rank. I want to show that $ (f^{*}\mathscr{E})^\vee\cong ...
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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, ...
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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 ...
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0answers
26 views

Show that the set of unitary matrices is not an affine algebraic variety in complex space $C^{n^2}$.

This is an exercise from An Invitation to Algebraic Geometry by Karen Smith. It asks to show that the set of unitary matrices $U_n$ is not an affine algebraic variety in complex space $C^{n^2}$. ...
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27 views

involution of a riemann surface

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if $\pi: C^{'}\rightarrow C$ it's a double unramified cover of a complex riemann surface named $C$ that ...
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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
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1answer
24 views

Number of connected components of a real variety

Let $f_1,\ldots,f_k\in\mathbb{R}[X_1,\ldots,X_n]$ with $d_i:=\deg f_i$ and suppose that $V:=\{x\in\mathbb{R}^n\,:\, f_1(x)=f_2(x)=\ldots=f_k(x)=0\}$ is of dimension $n-k$. I would like to bound the ...
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1answer
45 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 ...
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0answers
45 views

Theorem of Lefschetz

If anyone has the book of James D. Lewis entitled: A survey of Hodge conjecture on page $58$, There are the famous theorem of Lefschetz $(1,1)$ "without proof it seems to me." Is that so? Could you ...
2
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1answer
51 views

Let $f: U \rightarrow W$ be a morphism of affine algebraic sets and $f': k[W] \rightarrow k[U]$ be the k-algebra morphism of coordinate rings.

Prove if $f'$ is surjective then $f$ is a homeomorphism of $U$ onto the closed subset $W$. Well, it's the first time I've seen this word "homeomorphism" but I read online that a map is a ...
2
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1answer
30 views

Sheaf hom and the adjunction of push forward and inverse image

I'm trying to show that the tensor product of sheaves commutes with inverse image. I've reduced the problem to the following isomorphism $$f_*\mathscr{H}om_X(f^*\mathcal{N},\mathcal{P}) \cong ...
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1answer
27 views

Interaction of sheaf hom and push forward

I'm trying to show the following statement from this answer $$f_* \mathscr{H}om_X(A,\;B) \cong \mathscr{H}om_Y(f_* A,\; f_* B)$$ where $ f:X\rightarrow Y$ is a map of topological spaces and $ ...
2
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1answer
45 views

$\mathbb{A}^n$ with the Zariski Topology is Quasi-Compact.

I want to show that $\mathbb{A}^n$ is quasi-compact. I'm kind of stuck, I really don't know where to go with my proof, so I'll show what I have Proof: So suppose that $\cup U_i$ was an open cover for ...
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1answer
21 views

non-intersecting lines inside a projective quadric

In his book "Ideals, Varieties and Algorithms" D. Cox writes: Indeed, i can see that if $b \neq b'$ then $L_b$ does not intersect with $L_{b'}$. But does that not contradict the fact that two lines ...
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2answers
51 views

Global sections of the anticanonical bundle

Let $X$ be a smooth projective variety (over $\mathbb{C}$) with the canonical line bundle $K_X$. Also assume that $X$ has no global holomorphic top forms i.e. $H^0(X, K_X) = 0$. Is it true that the ...
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0answers
54 views

Is local homology group on a manifold a sheaf?

Let $X$ be a manifold of dimension $n$, and define $\mathcal{F}(U) = H_n(X,X-U)$. Then clearly $\mathcal{F}$ is a presheaf. I am thinking whether $\mathcal{F}$ is a sheaf. According to Lemma 3.27 in ...
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22 views

proof of Hartshorne on basic open sets of projective spectrum Proj S

In the proof of proposition 2.5 of Hartshorne's Algebraic Geometry, Chapter II, Section 2 it is written (somewhere in the middle): "The properties of localization show that $\phi$ is bijective as a ...
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0answers
52 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on ...
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2answers
56 views

Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$

There was an exercise I could not do. So the property is $T_1$ if for every pair of distinct points, $P, Q \in X$, there is an open subset $U$ containing $P$ but not $Q$ and another open subset $V$ ...
2
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1answer
29 views

Cubic hypersurface of singular conics

Conics in $\mathbb{P}^2$ are in one to one correspondence with points in $\mathbb{P}^5$, simple enough. Conics of rank one i.e. double lines are in a one to one correspondence with points on the ...