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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Verlinde formula, moduli space vector bundle on genus 2,3 curves.

I'd like to prove "by hands" the Verlinde formula for moduli space of rank two semistable vector bundles with fixed determinant on a curve of genus two and three. For a curve of genus two and even ...
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15 views

Isomorphism between $Pic_X[2]$ and $(\mathbb{Z}/2\mathbb{Z})^{2g}$

Let $X$ a Riemann surface of genus $g$. How could I prove that there is an isomorphism between $Pic_X[2]$ (i.e the line bundles $L$ such as $L^2 \simeq \mathcal{O}_X $) and ...
3
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1answer
49 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
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47 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
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22 views

Geometrical problem of maximum area of smallest triangle formed by 3 points in a distribution of n points on an R^2 plane

Another way of stating the gist of the question is: find the arrangement of n points such that one obtains the largest ratio between the area of the smallest triangle formed by three points to the ...
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1answer
57 views

Thinking About Fractional Ideals Geometrically

So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This ...
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21 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$. How is it possible to deduce that $G$ is conjugated to a subgroup of $GL_2(k)$ ?
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2answers
45 views

How to determine whether three ellipses have at least one common intersection point or not?

How to establish a criterion described in equation so that it is easy to determine whether three ellipses have common intersection area (point) or not? Update
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38 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
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1answer
54 views

How to determine if an equation is a rational variety?

Is there an easy way to determine if a given equation is a rational variety? For example, is $x^3+y^3+z^3=3$ a rational variety? I have maxima installed on my computer, if that helps.
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1answer
35 views

How to define a smooth subvariety as the vanishing of local coordinates

I keep stumbling upon this fact, and would like to see or get an idea for the proof: An ideal of a smooth subvariety at a point of a smooth variety can be generated by a subset of a suitably chosen ...
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1answer
32 views

Equivalent definitions for normal variety

Can anyone show (or provide a reference) that an algebraic variety $X$ is normal iff every finite, birational morphism $f:Y\rightarrow X$ is an isomorphism. More importantly, can you describe your ...
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39 views

Elliptic curves as $\mathbb{C}^*/\mathbb{Z}$

I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where ...
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1answer
65 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
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1answer
85 views

Why is $W_n(k)$ the unique flat lifting of a perfect field $k$ over $\mathbf{Z}/p^n$?

Let $k$ be a perfect field of characteristic $p>0$ and denote by $W_n(k)$ the ring of Witt vectors over $k$ of length $n$. In their article on the decomposition of the de Rham complex, Deligne and ...
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1answer
46 views

$\mathbb{P}(O\oplus O(-1))\simeq \mathbb{P}^1\times \mathbb{P}^1$?

Let $X=\mathbb{P}^1$. I am looking at $\mathbb{P}(O_X\oplus O_X(-1))$ and can see that it is the blow up of the projective plane at one point. I also see that it is a $\mathbb{P}^1$-bundle over $X$, ...
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2answers
115 views

Push forward of the structure sheaf along covering

Let $f: X \to Y$ be a covering (proper, surjective, finite regular map) of smooth projective varieties of degree $d$. How one can show that in this case $f_* \mathcal{O}_X$ is a locally free sheaf of ...
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37 views

Question about the Hodge conjecture

I would like to know how the mathematician : Mister Hodge define the map : $ \pi : C_k (X, \mathbb{Q} ) \to \mathrm{Hdg}_{k} ( X ) = H^{2k} ( X , \mathbb{Q} ) \bigcap H^{k,k} (X) $ which we need to ...
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0answers
20 views

definition about the generic rank of rational maps

I am reading a paper and there is a problem that says: Let $f:X\to Y$ a rational map, then the generic rank of $f$ is $k$. In particular, the dimension of the generic fiber of $f$ is $p$. I am not ...
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1answer
43 views

Twisting with a degree negative line bundle

Let $X$ be a Riemann surface. Let $M_1$ and $M_2$ be two holomorphic bundles on $X$. Does injectivity of $h^0(M_1)\to h^0(M_2)$ imply $h^0(M_1\otimes L)\to h^0(M_2\otimes L)$ is injective? Where $L$ ...
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0answers
41 views

what does it mean by k* here?

on the picture showing bijection, there is a notation k*. What does this stand for?
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0answers
30 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...
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2answers
41 views

How to show that this function is an isomorphisn??

Thank you all for helping me in the last question :) the question is how to prove that the 'natural map' from k to K is an isomorphism? I checked that it is injective, but can't show it is ...
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1answer
31 views

How does $F:Spec \ A \rightarrow Spec \ A'$ induce $F':A' \rightarrow A?$

I'm reading Algebraic Geometry by Ravi Vakil, and I'm trying to do exercise 4.3.A (the newest edition, I think June 2013). Show there is a bijection of the isomorphisms $\pi :Spec \ A \rightarrow Spec ...
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30 views

When is the map “attaching irreducible components” an effective isomorphism?

Let $\mathcal C$ be a category with fiber products. We say that a morphism $X \to Y$ in $\mathcal C$ is an effective epimorphism if the sequence of sets $$ \text{Hom}(Y,S) \to \text{Hom}(X,S) ...
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2answers
46 views

Proof of Noether's Normalization from Hulek's “Elementary Algebraic Geometry”

This is Klaus Hulek's Elementary Algebraic geometry. The star-checked part is the thing I can't understand. How can I eliminate all other $x_1, x_1', \ldots ,x_{n-1}, x_{n-1}'$ to make $f$ a ...
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1answer
39 views

Compute principal divisor for a rational function on a curve

During the lecture we defined the principal divisor of a rational function on a smooth curve as it follows: Consider the smooth curve $C\subseteq\mathbb{P}^2$. Take $g\in{K(C)^*}$. Then the principal ...
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61 views

Embedding an affine curve in a proper curve.

I am trying to figure out the following problem from Q.Liu's algebraic geometry in chapter 4. Let $U$ be an integral affine algebraic curve over $k$. (a) show that there exists a proper curve $ ...
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46 views

Could any one explain the difference between the theorems?

In the paper http://annals.math.princeton.edu/2007/165-2/p04 Theorem 2. Let $b \ge 2$ be an integer. The b-ary expansion of any irrational algebraic number cannot be generated by a finite automaton. ...
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1answer
42 views

Manifolds as homology classes

I have found that a k-dimensional submanifold of a manifold M can be considered as a class in the homology group $H_{k}(M)$. Why ?
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52 views

insight into the definition of intersection multiplicity for two plane curves

Let $X,Y$ be curves of $\mathbb{A}^2$ given by irreducible polynomials $f,g$ respectively, where the ground field $k$ is algebraically closed. Then it is known that the dimension of $k[x,y]/(f,g)$ as ...
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61 views

On the phrase “identify the graph…”

Let $\psi\colon \mathbb{P^1} \to \mathbb{P^1}$ be an isomorphism; identify the graph of $\psi$ as a subvariety of $\mathbb{P}^{1} \times \mathbb{P}^{1} \cong Q \subset \mathbb{P}^{3} $. Now do the ...
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0answers
39 views

Pushforward of a volume form

Let $X$ be a complex projective manifold with semi-ample line bundle $ K_X$ . Assume that $f: X\to X_{can}\subset \mathbb CP^N$ , and $f^{-1}(s)$ is nonsingular fibre, then I am looking for a proof ...
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0answers
37 views

Regular function on a variety which is not globally rational

I am looking for a particularly simple example of a regular function $f : V \to \mathbb{A}^1_k$ for some affine variety $V \subseteq \mathbb{A}^n_k$ over a field $k$, which cannot be expressed by a ...
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1answer
18 views

Birational map between manifolds

I have to show that the manifold $A=\{ [z_{1}:z_{2}:z_{3}:z_{4}]\in \mathbb{C}\mathbb{P}^{3} | z_{1}z_{3}^{n} - z_{2}z_{4}^{n}=0\}$ is birational equivalent to $\mathbb{C}\mathbb{P}^{2}$, how can I ...
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0answers
41 views

sections of birational proper morphism over an etale cover

Let $f: Y \to X$ be a birational proper morphism. Assume that every point of $X$ has an etale neighbourhood over which $f$ has a section. Is it true that $f$ is an isomorphism?
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1answer
44 views

Does “toric” conflict with “Calabi-Yau” in the projective case?

Let $X$ be a Calabi-Yau complex algebraic variety. If it is projective, we can talk about its geometric genus $p_g=h^{\dim X, 0}$, and the Calabi-Yau condition says that $p_g=1$. Now, one might be ...
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1answer
25 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
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2answers
44 views

question on quadric hypersurfaces

Over $\mathbb{C}$, every homogeneous polynomial of degree $2$ in $x_0,...,x_n$ can be brought into the form $f=x_0^2+...+x_r^2$ for some $0\le r\le n$. This is a part of an exercise of Hartshorne's ...
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0answers
15 views

Is there any bound on the number of generators of a monomial ideal in C(x,y)? [closed]

Just the question in the title, also if there is such a bound, say what it is.
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0answers
34 views

Is it important to study plane algebraic curves before read Fulton's book

I'm studying Fulton's algebraic curves book and I would like to know how important study plane algebraic geometry before read Fulton's book. Example of books on this subject: Algebraic Curves - ...
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2answers
52 views

Is there a general way to parameterize all implicit functions?

We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and ...
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1answer
31 views

What does linearly equivalent mean in this context

I'm trying to understand this proof of Fulton's algebraic curves book page 107: I didn't understand what does linearly equivalent mean in this context and why this implies it suffices to show that ...
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15 views

Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
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0answers
52 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
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1answer
48 views

Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...
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0answers
31 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
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1answer
33 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
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33 views

Scheme of Sections of a Coherent Sheaf

Suppose given a flat, projective morphism of finite type noetherian $\mathbb{C}$-schemes $X \rightarrow T$ and a coherent sheaf $M$ on $X$. Define a contravariant functor $F:Sch/T \rightarrow Grp$ ...
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38 views

Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...