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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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
26 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 ...
2
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1answer
33 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
30 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
56 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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1answer
100 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 ...
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1answer
55 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 ...
5
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1answer
87 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
54 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 $\mathcal{O}_{X,p}$ ...
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1answer
64 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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2answers
126 views

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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0answers
80 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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70 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 ...
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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 ...
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43 views

Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work ...
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1answer
23 views

Proving that a hypersurface is rational

Consider the algebraic set $V(F)$ in $\mathbb{A}^4_\mathbb{C}$, where $ F(x_1, x_2, x_3, x_4) = g(x_1, x_2, x_3, x_4) + h(x_1, x_2, x_3, x_4)$, $g(x_1, x_2, x_3, x_4)$ and $h(x_1, x_2, x_3, x_4)$ are ...
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1answer
36 views

Find the algebraic set $V(S)$

How can we find the algebraic set $$V(x^2+y^2-1)$$ ? $$V(S)=\{(a_1, a_2, \dots , a_n ) \in K^n |f_a(a_1, a_2, \dots , a_n )=0, \forall a \in A\}$$ where $$S=\{f_a \in K[x_1, x_2 , \dots , x_n] | a \in ...
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0answers
31 views

How to determine the acute angle bisector of given two lines

If we have two lines given by the equations: $$ax+by+c=0$$ $$px+qy+r=0$$ We know that the two angle bisectors are represented by the equations ...
2
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1answer
62 views

Jacobian of a Riemann surface and double unramified covers

Take the jacobian of a riemann surface named $J(C)$ as the set of the line bundle of degree zero. Set $J_{2}(C)$ the subgroup of $J(C)$ of the element of orther two i.e. $L\in J(C)$ such that ...
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0answers
41 views

On the proof that $k[\mathbb{A}^2\setminus\{0\}]=k[t_1,t_2]$?

I don't quite understand a proof that the coordinate ring of the punctured affine plane is $k[\mathbb{A}^2\setminus\{0\}]=k[t_1,t_2]$. The proof I have takes the principal open sets ...
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203 views

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 ...
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1answer
69 views

Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
5
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2answers
132 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
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0answers
66 views

(Soft Question) How active an area of research is Non-Commutative Geometry? [closed]

I am currently an undergraduate, but I am considering applying for a phd in algebraic geometry or a related field. I am quite interested in the link between non-commutative geometry and theoretical ...
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1answer
39 views

Counting parameters for the intersection of a quadric and a cubic surface in $\mathbb{P}^3$

I have to count parameters for the intersection of a quadric and a cubic surface in $\mathbb{P}^3$, up to linear automorphisms of $\mathbb{P}^3$. I take account of the theorem according to which a not ...
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1answer
43 views

Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes

I am reading the proof by Götz-Wedhorn Algebraic Geometry I Theorem 6.45, and also Liu, Theorem 1.14. One thing that I do not understand is this: For easier cases, we assume $X=\text{Spec A}$ and let ...
3
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1answer
41 views

Fixed point subspaces $V^B$ and $V^G$ for a Borel subgroup $B\subset G$ coincide

Assume that $G$ is a linear algebraic group, and let $B \subset G$ a Borel subgroup of it. Let $(V,\rho)$ a rational $G$-module. Define $$V^G := \{ v \in V \mid g \cdot v = v \quad \forall g \in G ...
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1answer
62 views

Properties of fibers of a morphism of varieties

In this question, all varieties are supposed to be over an algebraically closed field $k$. Hypothesis: X is a smooth projective surface and $f:X\longrightarrow \mathbb P^1$ is a morphism with we ...
3
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1answer
47 views

Embeddings into products of projective spaces and multi graded rings

Let X be a variety over a field $k$. Assume that it is embedded into product of two projective spaces $$ i : X \subset \mathbb{P}^n \times \mathbb{P}^m. $$ I want to construct a graded algebra that ...
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52 views

Stalk of a locally finite type $k$-scheme, where $k$ is a field.

I think there is something I am not understanding and I am a bit confused at the moment. I would appreciate any help! Let $X$ be a locally finite type $k$-scheme, where $k$ is a field. Say $p \in ...
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1answer
193 views

What's the “real” reason a finite map has finite fibers?

This is a soft question. I have encountered two very different proofs of what seems like "basically the same theorem," and I want to understand how they relate and "what the real explanation is." ...
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1answer
37 views

closed point of a locally finite $k$-scheme

Let $X$ be a locally finite $k$-scheme, where $k$ is a field. Suppose I have $Spec B \subseteq X$ such that $B$ is a finitely generated $k$-algebra, and $p \in Spec B$ a closed point inside $Spec B$ ...
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1answer
23 views

Dimension of the intersection between a projective variety and a hyperplane.

Suppose that $X$ is a smooth $m$-dimensional projective variety embedded in some $\mathbb P^n_k$ (we work over an algebraically closed field). Now consider a hyperplane $H\subseteq\mathbb P^n_k$ of ...
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45 views

Characterization of the parametrizing spaces of quadrics

Consider the quadrics in $\mathbb{P}^3$. They are parametrized by the points of $\mathbb{P}^9$ and the space parametrizing the quadrics of rank 1 (by the rank of a quadric I mean the rank of the ...
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1answer
25 views

the point is outside or on the line perimeter?

How do I calculate if a point is inside or outside the perimeter, between two points as shown in the picture? The point A and point B have a perimeter R. How do I know if the point C is on the ...
1
vote
1answer
43 views

equivalence of two notions of linear subspaces in general position

Let $V_1,\dots,V_n$ be linear subspaces of $k^D$, where $k$ is a field. For any subset $S$ of $[n]:=\left\{1,\dots,n\right\}$ define $V_S = \sum_{i \in S} V_i$ and $W_S = \bigcap_{i \in S} V_i$. ...