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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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 ...
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61 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
23 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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40 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 ...
2
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1answer
59 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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57 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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36 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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39 views

how would i answer this question [closed]

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
47 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
24 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
63 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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36 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
25 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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33 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
38 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
48 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.
2
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1answer
57 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
41 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 ...
2
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2answers
45 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
47 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
47 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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27 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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30 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 ...
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1answer
25 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
53 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
47 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
54 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 ...
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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
55 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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55 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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23 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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74 views
+50

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
58 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 $V$ ...
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1answer
31 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 ...
3
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1answer
54 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 ...
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1answer
86 views

In which cases does pullback commute with the Hom-sheaf?

Assume $f: (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a morphism of locally ringed spaces and E and F are two locally free $\mathcal{O}_Y$-moduels of finite rank. I was wondering if we have ...
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1answer
34 views

Self-intersection of a cycle

If $X$ is a smooth projective variety of dimension $2n$, and $V \subset X$ is a smooth subvariety of dimension $\geq n$, then $V \cdot V$ makes sense as a class as an element of the Chow ring ...
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1answer
66 views

Isomorphic elliptic curves are projectively equivalent

Let $E_1$, $E_2 \subset \mathbb{P}^d$ be two smooth elliptic curves, that are isomorphic as abstract curves. How can one prove that they are projectively equivalent? That is there is a automorphism ...
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1answer
53 views

Closed points and discrete valuation rings, Exercise in Ravi Vakil's notes 12.7.B

I am working on the problem in Vakil's notes exercise 12.7.B which asks Suppose $X$ is an irreducible Noetherian separated curve. If $p\in X$ is a regular closed point, then $\mathcal{O}_{X,p}$ ...
3
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1answer
62 views

Riemann-Hurwitz formula generalization in higher dimension

In "Basic algebraic geometry 2", Shafarevich finds a relation between the Euler characteristic and the genus of the curve. At page 139 he says that there's no analogue for varieties of dimension ...
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2answers
36 views

What is the value of $k$ in the plane $x + ky = 1$ such that the intersection of the plane and hyperboloid $y^2 - x^2 - z^2 = 1$ is ellipsoid? [closed]

What is the value of $k$ in the equation of plane $x + k\;y = 1$ such that the intersection of the plane and hyperboloid $y^2 - x^2 - z^2 = 1$ is an ellipse? I tried to use substitution method but it ...
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105 views
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Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
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79 views

A future in mathematics

I am a Junior in high school right now, trying to figure out what to do next mathematically. I have familiarity with real analysis (Baby Rudin, and also a bit on the gauge integral), complex analysis ...
2
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1answer
38 views

Morphism $f\colon X\to S$ is proper iff $f^{-1}(V_j)\to V_j$ is proper for some open cover $\{V_j\}$ of $S$? (Lemma 28.42.3 of Stacks Project)

I was browsing the Stacks Project, and Lemma 28.42.3 says that a morphism $f\colon X\to S$ is a proper morphism if and only if there exists an open covering $S=\bigcup V_j$ such that $f^{-1}(V_j)\to ...
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68 views

Contraction of curves on a surface and stalks

Let $$f:X \rightarrow S$$ be a fibered surface over a Dedekind scheme of dimension $1.$ Let $$s_1, \ldots, s_n$$ be closed points of S and $\{E_{ij}\}$ irreducible vertical divisors of $X$ with ...
2
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1answer
28 views

What is $\{ f \in F_{p^m}[x_1, \ldots, x_n] : f(a) = 0, \forall a \in A^n\}$?

As the title suggests, I am interested in knowing if there is a neat description of the ideal of polynomials that vanish on affine $n$ space over a finite field with $p^m$ elements. Is there a way to ...