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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28 views

One-One correspondence between elements of G (algebraic group) and maximal ideals of K[G]

We are given that $G$ is an algebraic group over $K$ and $K[G]= K[x_1,...,x_n]/I(G)$ where $I(G)$ is the ideal consisting of all polynomials of which elements of $G$ are the common zeroes. Now,if we ...
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1answer
29 views

Why do we need to take the closure in the definition of projective closure?

Let $X=Z(I)$ be an algebraic set in $\mathbb{A}^n$. Given the standard covering $\{U_i\}$ of $\mathbb{P}^n$ and the homeomorphism $\varphi_0:U_0\to\mathbb{A}^n$, the projective closure of $X$ is ...
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31 views

Groebner basis and projective closures

New to algebraic geometry and Groebner basis, so I just wanted to bounce my argument off of somebody. I have a zero set defined by one polynomial, $Y=Z(S) = \{p(x)\}$ in affine space, I am interested ...
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11 views

Non-convex subdivisions of newton polygon of a tropical plane curve

This is probably an elementary question, but how come the Newton polygon of a tropical plane curve can't have non-convex subdivisions? Or can it?
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60 views

Intersection condition for a Grothendieck topology

I am a bit confused about constructing a Grothendieck topology from a Grothendieck pretopology, largely because I have a discrepancy in definitions of the former. According to all of the questions ...
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35 views

Jacobian criterion algebraic independence

I have two polynomials $f_{1}(x,y)$ and $f_{2}(x,y)$ and I want to know if they are algebraically independent. I am using the Jacobian criterion which says that $f_{1}$ and $f_{2}$ are algebraically ...
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35 views

Why are none of the $\overline{Y_i}$ contained in one another?

Let $X$ be a topological space which is a union of finitely many irreducible closed sets $X_1, ... , X_n$. Lemma: if none of the $X_i$ are contained in one another, then $X_1, ... , X_n$ are the ...
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2answers
48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [on hold]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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1answer
43 views

Example of variety with big automorphism group? (Big as in, not a variety in a reasonable way.)

Let $X$ be some algebraic variety over $k$. In some situations, like $X = P^1_k$ (where it some quasi-projective variety), the set of automorphisms has a natural algebraic structure. (If $X$ is a ...
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38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
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1answer
102 views

Grothendieck's “Relative” Point of View

I have often read that Grothendieck's insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why ...
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2answers
44 views

Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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1answer
47 views

Algebraic K-theory of the cotangent bundle

Below, always let $A$ be the coordinate ring of a smooth affine variety over $\mathbb C$. What can be said about the (non)-triviality of the module of Kahler differentials $\Omega_{A/\mathbb C}^1$? ...
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56 views

Why is the identity function not the exponential of a holomorphic function on $\mathbb{C}\backslash\{0\}$?

In chapter 2 of Qing Liu's book Algebraic Geometry and Arithmetic Curves he states that the identity function on $X=\mathbb{C}\backslash\{0\}$ is not the exponential of a holomorphic function on $X$. ...
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1answer
41 views

If $Y_i$ are the irr. components of $Y$, then $\overline{Y_i}$ are the irr. componens of $\overline{Y}$.

Got a real dumb question for ya. Suppose $Y$ is a subset of a topological space with irreducible components $Y_1, ... , Y_n$. Then $\overline{Y_1}, ... , \overline{Y_n}$ should be the irreducible ...
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2answers
41 views

Length of the intersection of two quadric surfaces.

How do I find the length of the intersection between $$ax^2+(a+1)z^2=1$$ and $$x^2+y^2+z^2=\dfrac{1}{a}\ ?$$ The idea is to make a parametrization of the curves using the variable $t$: $$x = ...
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107 views

Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a ...
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1answer
52 views

Hartshorne proposition 1.2 e)

We want to prove that $ Z(I(Y)) \subseteq \overline Y$. Let $W$ be any closed set containing $Y$. Then $W=Z(a)$ for some ideal $a$. So $Y \subseteq Z(a)$ and $I(Z(a)) \subseteq I(Y)$. Clearly we have ...
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1answer
25 views

Order of Contact for a general tangent line of a cubic threefold

I am trying to solve Exercise 18.21 of Harris "Algebraic Geometry". In the proof of the unirationality of a smooth cubic threefold X he claims that a general tangent line to X at a general point p ...
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1answer
46 views

Irreducibility of universal hyperplane section

In this question, Georges Elencwajg gives a fantastic geometric answer. However, this answer rests on the fact that the universal hyperplane section $\Omega_X$ is irreducible. I only know how to prove ...
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3answers
6k views

Grothendieck 's question - any update?

I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
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1answer
82 views

Induced ring homomorphism by map on spectra

I know that when we have a ring homomorphism $$ \phi: A\rightarrow B$$ this induces continuous map between the spectra $$ \phi': \operatorname{Spec} B\rightarrow \operatorname{Spec}A$$ which maps $p$ ...
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2answers
60 views

A non-zero section of an invertible sheaf on a geometrically integral (smooth?) projective $k$-scheme is regular?

Let $X$ be a projective, geometrically integral $k$-scheme $X$ for a field $k$ (possibly we also need $X$ smooth, i.e. $X_{\overline{k}}$ regular). It seems implicit in Hartshorne's discussion (with ...
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1answer
64 views

Is there an elementary way to prove the set of fibers containing a variety is closed?

Let $X\subset \mathbb P^n$ be a projective variety, $B$ a projective variety, and $V\subset B\times \mathbb P^n$ a family over $B$. Denote the fiber over a point by $V_b$. Exercise 4.4 in Harris's ...
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29 views

Class group of the cone

This is from exercise 6.3a of Hartshorne. Let $V$ be a projective variety over a field $k$ of dimension $\geq 1$ that is non-singular in codimension 1. Let $X = C(V)$ be the affine cone over $V$ in ...
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0answers
24 views

Order of contact for a general tangent line of a cubic hypersurface

For the polynomial $y=x^{3}$ it is easy to see, that for every point except $x=0$ the order of contact for a tangent line is 2. Harris uses in his book "Algebraic geometry" the fact, that a general ...
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2answers
71 views

Is an elliptic curve canonically isomorphic to its quotient by its own $n$-torsion?

Let $E$ be an elliptic curve (say, over a field $K$), and $E[n]$ be its $n$-torsion subgroup-scheme (suppose char $K$ is coprime to $n$). Is $E$ canonically isomorphic to $E/E[n]$? (What is this ...
3
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1answer
55 views

Smooth affine plane curve with non-trivial cotangent sheaf?

Question: Let $A = \mathbb C[x,y]/(f)$ be a non-singular plane curve. Under what conditions is the module of Kahler differentials $\Omega_A^1$ (over $\mathbb C$) a free module? I am not sure what ...
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41 views

Irreducibility of the universal hypersurface family

Fix $d>1$. Suppose $X\subset \mathbb P^N\times\mathbb P^n$ is the universal family of degree $d$ hypersurfaces in $\mathbb P^n$. I would like to show this does not admit a rational section $\sigma: ...
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1answer
41 views

Lie Algebra of a connected simple linear algebraic group

Let $G$ be a linear algebraic group and $A=K[G]$ (K is a field of characterstic 0) be the coordinate ring of $G$. In Humphreys, the Lie algebra of $G$ is defined as the space of left invariant ...
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53 views

Example of a variety whose image by polynomial maps is not a variety

If $X$ is an algebraic variety over a field $k$, and $G$ acts by polynomial maps, one might hope that $X/G$ could be made into an algebraic variety Could someone give an example of a variety in ...
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1answer
74 views

About rational Hodge conjecture.

What progress has been made to date on the rational Hodge conjecture ? Can anyone tell us if there is some new books related to Hodge conjecture which explain in detail, the latest development in the ...
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2answers
147 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
3
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1answer
64 views

What is the definition of a presheaf in EGA?

In EGA I, Grothendieck says he is not going to bother recalling the definition of a presheaf (on a given topological space $X$ with values in some category $\textbf{K}$). I was just wondering what ...
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28 views

Branch point of hyperelliptic Riemann surface

I am now learning compact Riemann surface, could someone help me with the following problem? $X$ is hyperelliptic Riemann Surface with genus $g$, let $P$ be the branching point, then $\dim L(P)=1$ ...
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1answer
35 views

Vector bundle morphisms and determinants

Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$. Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is ...
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41 views

Another computation of the $Ext^1$ sheaf on a surface

Let $X$ be a smooth surface and $A$ a line bundle on a smooth curve $C\subset X$, which we view as a torsion sheaf on $X$. By taking $Hom(A,-)$ of the sequence $0\to \mathcal{O}_X\to ...
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1answer
43 views

Is every integral point of an arithmetic scheme contained in an affine open set?

The schemes $$X = Proj \mathbb{Z}[s,t] = \mathbb{P}^1_{\mathbb{Z}}$$ and $$Y = Proj \mathbb{Z}[x,y,z]/(x^2 + y^2 - z^2)$$ both have isomorphic generic fibers as schemes over $\mathbb{Z}$, and there is ...
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1answer
32 views

Given a (2,1) category is there a canonical way of constructing a 1-category?

Given a (2,1)-category (for example, the category of algebraic stacks over some fixed site), you can consider the 1-category with the same objects, but where morphisms between the objects are just ...
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26 views

Is multiplicative inverse defined for ideal? Eg. $x^3 y\in \langle x^3 y^2\rangle$?

Definition. A subset $I\subset k[x_1,\ldots,x_n]$ is an ideal if i. $0\in I.$ ii. If $f,g\in I$, then $f+g\in I$. iii. If $f\in I$ and $h\in k[x_1,\ldots,x_n]$, then $hf\in I$. ...
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1answer
37 views

Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
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13 views

Fiber product of toric varieties

Let $X$,$Y$ and $Z$ be three toric varieties defined by the fans $\Sigma_X\subset (N_X)_{\mathbb{R}}$, $\Sigma_Y\subset (N_Y)_{\mathbb{R}}$ and $\Sigma_Z\subset (N_Z)_{\mathbb{R}}$, respectively. It ...
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0answers
34 views

$A_r$-singularities and effectiveness of some divisor.

Let $X=\text{Spec}(\mathbb{C}[x,y,t]\big/(xy-t^n))$ for some $n$. Let $Y\subset X$ be defined by the prime ideal $(y,t)$. If we know that $$K_X-mY$$ is effective, can we say something about the ...
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26 views

Difficulty with Milnor number

I am reading through the wikipedia page of Milnor number : https://en.wikipedia.org/wiki/Milnor_number#Examples I am reading example 2 where they calculate the Milnor number of $f(x,y)=x^3+xy^2$. So ...
3
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1answer
46 views

Equivalent condition for a linear map to coincide with another restricted to a line modulo certain hyperplane.

Let $p$ be a line of the vector space $K^{n+1}$, and let $H$ be a hyperplane of $K^{n+1}$ such that $p\subseteq H$. We may also interpretate $H$ as a line $q\subseteq(K^{n+1})^{*}$. Let ...
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1answer
53 views

Trying to understand some basic facts about tangent space of Grassmannian.

I am reading Harris's 'Algebraic Geometry: A first course'. I am trying to understand its identification of the tangent space to a Grassmannian. Let $G(k,n)$ be the Grassmannian of $k$-planes in ...
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13 views

Geometric intuition for conjunctive spaces

A topological space $S$ will be called conjunctive if for each open set $A$ containing a point $p$, there's a point $q\in S$ satisfying $\overline{\left\{q \right\}}\subset A\cap \overline{\left\{p ...
3
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1answer
106 views

Cannonical evaluation map

Let $C$ be a curve over $\mathbb{C}$, and $E$ be a vector bundle on $C$ such that $H^0 (C, E) \neq 0$. Everyone talks of the evaluation map $H^0 (C, E)\otimes O_C\longrightarrow E$. What is this map ...
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1answer
140 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 ...
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1answer
75 views

An inequality about the dimension of fiber

I am working on Problem 11.4.A of Vakil's notes: Let $X$ and $Y$ be two locally noetherian schemes, and $\pi:X \to Y$ is a morphism. $\pi(p)=q$. Then prove: $codim_Xp \leq ...