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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Question to understand how sections work

Suppose that $C$ is a compact Riemann surface o positive genus $g$. Let $L$ a linear bundle on $C$ . Chosen a divisor $D$ on $C$ we consider the set $L(D)$ that is the tensor product between $L$ and ...
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3answers
159 views

Are there such things as 'locally homogenous spaces'?

A Euclidean space has the property that every point has a neighbourhood that is homeomorphic to some neighbourhood of any other point. I'm not sure what the name of this property is - I thought it ...
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0answers
17 views

The map $x \mapsto gx$ gives a homeomorphism $G \rightarrow G$ for algebraic groups.

Let $G$ be an algebraic group. The product group $G \times G$ (taken as a product of varieties) contains the product topology, and is a product with respect to the canonical projections $G \times G ...
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1answer
126 views

What's the intution behind defining the cotangent sheaf as $\Delta^\ast(\mathscr{I}/\mathscr{I}^2)$?

This definition seems to be given all over the place (e.g. Hartshorne II.8, Vakil 21.2.20, Wikipedia, McKernan's lecture notes from MIT), and never with any explanation as to why the map $\Delta : X ...
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1answer
22 views

Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Everything is complex algebraic. Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where ...
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24 views

Determine whether a regular surjective map is finite

Consider the regular map between affine closed sets $f \colon \mathbb{A}^1 \rightarrow \mathcal{Z}(y^2-x^3) \subseteq \mathbb{A}^2$ given by $f(t) = (t^2,t^3)$. $f$ is obviously a dominant map. I ...
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49 views

How local is the exponent in the definition of a function with analytic/algebraic singularities?

In Demailly's Analytic Methods in Algebraic Geometry (available on his web page), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: ...
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35 views

Analogue of splitting field in several variables

Let $k$ be a field, and $P \in k[X]$. Consider the extensions $k \subset L \subset K$, where $L$ is a splitting field for $P$ over $k$ and $K$ is the algebraic closure of $k$. Then (by definition) all ...
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0answers
34 views

Fibres of the base change of a scheme

I am trying to gain a better understanding of the notion of fibre products of schemes. Two major applications that I've began to study are: 1) Making an $S$-scheme $X$ into an $S'$-scheme via a ...
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1answer
66 views

What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in ...
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1answer
47 views

When is the tensor product of a separable field extension with itself a domain?

I'm reading Algebraic Geometry and Arithmetic Curves by Qing Liu. On page 92, in the proof of Corollary 3.2.14 d), he states that if $K \otimes_k K$ is a domain, then $K = k$. Here $K$ is a separable ...
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47 views

A question on GCT

In http://ramakrishnadas.cs.uchicago.edu/gctriemann.ps it is stated that there is an unknown non-standard riemann hypothesis. AFAIK riemann hypothesis in AG was shown using Etale cohomology by Artin, ...
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1answer
66 views

Does dominant morphism of integral schemes is injective on sheaves?

Let $f:X \to Y$ be a dominant morphism of integral schemes. Is it true that it is equivalent to the fact that $\mathcal O_Y \to f_* \mathcal O_X$ is injective? Or does one imply another? It's quite ...
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1answer
84 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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1answer
71 views

Did I use axiom of choice in my proof?

I have two different affine open covers for a scheme $X$, say $X = \cup_{i \in I} U_i$ and $X = \cup_{j \in J} V_j$. For each $p \in X$, we know there exist some $i(p)$ and $j(p)$ such that $p \in ...
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1answer
531 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
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31 views

Set-theoretic intersection of affine open subschemes.

Let $X$ be a separated scheme, $U,V \subseteq X$ open affine subschemes, $\Delta \colon X \to X \times X$ the diagonal morphism and $\pi_1, \pi_2 \colon X \times X \to X $ the natural projections, so ...
3
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0answers
43 views

Locus where morphism is étale is open

Let $f : X \to Y$ be a morphism of schemes, flat, finite and locally of finite presentation of rank $d$. Is it true that the locus $$\{ y \in Y : f_y : X_y \to y \hspace{1mm} \text{is étale}\}$$ is ...
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1answer
42 views

Universal Property of the Universal Line

In "An Invitation to Quantum Cohomology" by Kock and Vainsencher, they talk about "the universal line", which is defined as the variety $U=\{ (L,p)\in Gr(1,\mathbb{P}^r)\times \mathbb{P}^r | p\in L ...
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1answer
25 views

Circle passing through intersection points of two bigger circles

Suppose the equations of two intersecting circles are given.Now how to find the equation of circle passing through the points of intersection of the larger circles? Now please dont tell me that i got ...
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32 views

Some questions about tropical geometry: graphs of tropical curves.

I am reading the file about tropical geometry. I have some questions about the file. The questions are in the following. On page 33 of the file, why the tropical version of $$ 0.001 + 1000 x + 100 ...
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0answers
13 views

Arithmetic picard rank of smooth cubic surfaces

Assume a smooth cubic surface is defined over a field $k$ characteristic $0$, that it has line defined over $k$ and that its arithmetic Picard rank over $k$ is maximal i.e. $7$. Does this imply that ...
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1answer
36 views

Product of affine varieties is the product of topological spaces

Let $k$ be an algebraically closed field, and $A, B$ affine $k$-algebras. We can define a functor $\mathfrak F$ from the category of affine $k$-algebras to that of affine algebraic varieties, by ...
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30 views

Direct image of the structure sheaf of $X\times\mathbb{P}^1$

Let $X$ be a projective variety, $Y=X\times\mathbb{P}^1$ and $p:Y\to X$ be a projection. Since the dimension of fibres of $p$ is equal to one we have $R^ip_*\mathcal{O}_Y=0$ for $i>1$. Is it true ...
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1answer
32 views

Help with a definition on algebraic geometry

Can anybody give me a reference to understand the definition of "a smooth complete integral pointed algebraic curve"? I'm beginning to study the paper "upper bounds for the dimension of moduli spaces ...
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0answers
36 views

Can anyone comment on uniformizing parameters and uniformizing coordinates?

Let $V$ - an algebraic variety ($\dim V = r$), $U \subseteq V$ - an open subset (in Zariski topology), P - a prime divisor of V, that is, the closed subvariety such that $\mathrm{codim}_VP = 1$. On ...
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1answer
24 views

Based on the Andreotti-Frankel theorem, what is the CW complex homotopy equivalent to $x^2 + y^2 - 1$?

I am referring to this theorem: http://en.wikipedia.org/wiki/Andreotti%E2%80%93Frankel_theorem I have no idea how to begin thinking about this.
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0answers
25 views

Question about a notation of line bundle

If $C$ is a complex Riemann Surface with positive genus, $D$ a divisor on $C$, $L$ a line bundle of $C$, with the term $L(D)$ what do we mean? I have this idea: $L(D)$ set of all sections of $L$ ...
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1answer
32 views

Lines on a singular cubic surface

How many lines the cubic surfaces $xyz=w^3 \in \mathbb P^3$ has? I found only three: $x=w=0$, $y=w=0$ and $z=w=0$. How to prove that there are no other lines? Also, this surface is singular, is it ...
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1answer
48 views

The higher cohomologies of a quasi-coherent sheaf on the intersection of two affine open subsets.

It is well-known in algebraic geometry that if $X$ is affine and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $$ H^i(X,\mathcal{F})=0,~ \forall ~i\geq 1. $$ Now let $X$ be an arbitrary ...
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1answer
57 views
+50

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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1answer
80 views
+50

poles and zeros of function field of $\mathbb{P}^1$.

In which condition: an element of function field of $\mathbb{P}^1$ has zero or pole or no-zero&no-pole. I am thinking that: since $\mathbb{P}^1$ and $\mathbb{A}^1$ is birrationally equivalent ...
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5answers
725 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
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0answers
33 views

Techniques of Sobolev spaces in algebraic geometry

I learned a proof of Hodge decomposition (in particular harmonic integral) in Griffith and Harris's book. It was proven by techniques of Sobolev spaces. But I wonder when I use techniques of this type ...
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0answers
25 views

Is the push-forwad of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
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1answer
63 views

Irreducible components of schemes

Consider the scheme $X:=\mathrm{Spec}(k[X,Y]/(X^2,XY))$. According to Qing Liu's "Algebraic geometry and arithmetic curves", the irreducible components are in $1-1$ correspondence with subschemes of ...
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38 views

Inverse image of line bundle under blowing up

Let $X$ be a smooth projective manifold, $D$ be a divisor on $X$ and $\pi:Y\to X$ be a blowing up with a smooth centre. Is it true that $\pi^*(\mathcal{O}_X(D))\cong\mathcal{O}_Y(\pi^*D)$?
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1answer
45 views

Succesive vs simultaneous blow-up

I'm studying blow-ups of varieties and there's some things which seems confusing to me. Denote by $Z={\rm Bl}_{x_1,x_2}(X)$ the blow-up of two points $x_1, x_2\in X$. Under what conditions $Z$ is ...
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1answer
34 views

Properties of dominant morphism of schemes

I am trying to solve the following exercise 4.11, p.67 from Qing Liu's book "Algebraic geometry and arithmetic curves". Let $f:X\to Y$ be a morphism of irreducible schemes with respective generic ...
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0answers
31 views

Question about definition of a regular point

It says on Wikipedia: A general algebraic variety V being defined as the common zeros of several polynomials, the condition on a point P of V to be singular point is that the Jacobian matrix of the ...
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0answers
40 views

Hartshorne Exercise V.1.8 - On NumX

This exercise is intended to show that Num$X$ (divisors modulo numerical equivalence) on a smooth, projective, integral surface $X$ is a finitely generated free abelian group, without using some big ...
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33 views

Equivalence of definitions for completion

For the settings on my question, take Atiyah's chapter on completions. Basically we have two definitions of completness (Atiyah's sense, the canonical map $\phi:M\rightarrow \widehat{M}$ is an ...
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2answers
40 views

any two sets of $n+2$ points are projectively equivalent in $\mathbb{P}^n$

Problem: Any two sets of $n+2$ points in general position in $\mathbb{P}^n$ are projectively equivalent. In thinking about this problem, it is natural for me to reduce it to linear algebra ...
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0answers
27 views

Algebraic Curves, Fulton, Exercise 2.22: affine change of coordinates and isomorphisms between local rings

I'm struggling with the following exercise I need to solve and present in our Algebraic Geometry class (book: Algebraic Curves, William Fulton, 2008 Edition, Exercise 2.22): Let $k$ be an ...
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1answer
43 views

Algebraic Geometry Approach To Study The Surfaces Given The Intersection Curve

I'm NOT a Mathematician and I'm totally new to the field of Algebraic Geometry. A friend of mine told me that one thing which is studied in this field is to consider a curve as a set of points in n-D ...
3
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0answers
46 views

What is the moduli space of Calabi-Yau manifolds

This is supposed to be a generic question for Calabi-Yau manifolds, but for definiteness let me exemplify it with the $K3$ manifold. What is the moduli space of $K3$ manifolds? I am also asking what ...
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0answers
37 views

Dimension of a Vector Space of section

Let $C$ a compact Riemann Surface with genus $g>0$ and $L$ a theta characteristic on $C$ that is $L^{\otimes2} \equiv \omega_C$ where $\omega_C$ is the canonical bundle of $C$. Take on $C$ the ...
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3answers
86 views

There is a unique quadric through three disjoint lines

There is a classical exercise that three disjoint lines in $\mathbb{P}^3$ are contained in a quadric surface $Q$. The existence is trivial. Every quadric in $\mathbb{P}^3$ is determined by nine ...
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0answers
118 views
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Higher dimensional analogue for Riemann Hurwitz formula

There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and ...
3
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1answer
66 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...