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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18 views

Hypotheses for Leray's theorem.

Why is it not enough, in the hypotheses for Leray's theorem, to assume that $H^q(U_a, \mathcal{F}) = 0$, for all $q > 0$?
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1answer
19 views

Inclusion of rings and induced map, fibers?

Consider the inclusion of rings $\mathbb{R}[X] \subset \mathbb{C}[X]$ and the corresponding induced map $\phi: \text{Spec}\,\mathbb{C}[x] \to \text{Spec}\,\mathbb{R}[X]$. Can someone give me an ...
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2answers
40 views

coefficients that make $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ a constant

Find integers $a_i$ and $b_i$, $i=1,2,3,4$, such that $p(x)$ is a constant function: $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ I don't even know if such coefficients exist or not. ...
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20 views

Normal irreducible quadrics in $\mathbb{P}^3$

How can I show that an irreducible quadric $Q$ in $\mathbb{P}^3$ is normal? If $Q$ is non singular then the local ring associated to every point is a UFD and so a normal ring, but how can I do ...
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24 views

Is it true that $\mathrm{codim}(Z \cap Y, Y) \leq\mathrm{codim}(Z,X)$ for closed subsets $Z,Y$ of a scheme $X$?

This might be a standard thing but I'm not so sure. Say $X$ is an irreducible affine scheme, $Y$ is an irreducible closed subset of $X$, and $Z$ a closed subset of $X$. If $Z \cap Y \neq ...
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23 views

“Every regular function on an affine variety is polynomial” - generalisation to the case of a reducible variety

$K$ is an algebraically-closed field, and for an affine variety $X$, $A(X) $ denotes the ring of polynomial functions on $X$. What I would like to prove is the following: "Let $X \subset ...
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22 views

To get a curve on the ample linear system

Let $X$ be a surface which has finitely many nodes. Let $f:Y\longrightarrow X$ be the desingularization of $X$, that is the blow up of $X$ at these finitely many points. Given an ample vector bundle ...
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0answers
27 views

Grothendieck spectral sequence from the hypercohomology spectral sequence

Is it possible to write a proof of the convergence of the Grothendieck spectral sequence of the composition of two functors only using the convergence of the hypercohomology spectral sequences ...
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20 views

intersection of plane elliptic curve with coordinate hyperplanes

Let $E: y^2z = x^3 - Axz^2 - Bz^3$ be a plane elliptic curve. I want to calculate the intersection of $E$ with the coordinate hyperplanes $H_i = \{x_i = 0\}$, $i=1,2,3$. I write $H_x = \{x=0\}, H_y = ...
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20 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
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30 views

Definition of $K$-linear

What does it mean for an element to be $k$-linear, where $k$ is a field? Or what does it mean that an arbitrary polynomial $f \in \mathbb{R}[x,y,z]$ is an $\mathbb{R}$-linear combination of monomials? ...
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44 views

Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
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1answer
37 views

Degree of a projected curve

I cannot find the proof of the following fact, can anyone help me? Let $C$ be a projective curve in $P^n$ and $p\in C$ a smooth point. Let $C'$ be the closure in $P^{n-1}$ of the image of C\p via the ...
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187 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
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0answers
27 views

A computation of a dimension of a space of sections

Let $S$ be a finite set with $m$ elements in $\mathbb{P}^n$. Associate to $S$ the subscheme $Y$ such that the ideal sheaf $\mathcal{I}_Y$ of $Y$ is the ideal sheaf of $S$ to the power $t$, where ...
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0answers
28 views

Dimension of cohomology group of sheaves associated to a point: $dimH^0(L(−P))=dimH^0(L)−1$

Could anybody help me with this theorem? Let $L$ be a line bundle on a smooth projective curve with $H^0(L)$ positive dimensional, then for a general point P, $dimH^0(L(−P))=dimH^0(L)−1$. I don't ...
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44 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
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39 views

Is the pull back of the curve in the linear system?

Let $X$ be a surface. And let $f:Y\longrightarrow X$ be the blow up $X$ at finitely many points. Let $L$ be an ample line bundle on $X$ and let $C\in |L|$ be a smooth curve on $X$ which avoids the ...
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1answer
29 views

Does flatness imply components map dominantly?

I suspect the following is well-known, but I cannot find a reference. Let $f:X\to Y$ be a morphism of complex algebraic schemes, which is flat. We can assume both $X$ and $Y$ are reduced. Is it true ...
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19 views

Radicial Morphism over DVR's

I would like a reference for the truth/falsity of the following statement: Suppose that $X \rightarrow Y$ is a map of $S$ schemes where $S$ is the spectrum of a DVR with generic point $\eta$ and ...
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0answers
28 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
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35 views

When does a principal divisor have degree 0?

We know that on a smooth curve over an algebraically closed field principal divisors have degree 0. This holds true also for the projective space over any field (in this case equality holds too). My ...
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30 views

Codimension of Schemes

Let $X$ be an integral scheme over a field $k$ of finite type. For a closed non-empty subset $Y$ show that $\operatorname{codim}(Y,X) = \inf \{ \dim \mathcal{O}_y \mid y\in Y \}$. It is easy to prove ...
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24 views

Constructing relatively ample line bundles

Let us work over $\mathbb C$. Let $X$ be a smooth projective variety with an ample line bundle $L$. Let $S$ be any scheme over $\mathbb C$, reduced or integral (or... add more assumptions if ...
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1answer
40 views

Formal smoothness of $A \to A[t]/(h)$.

Let $A$ be a commutative noetherian ring, $T$ an indeterminate, $h=h(T) \in A[T]$, and $B:= A[T]/(h)$. When $B$ is formally smooth over $A$? (If $h$ is monic, is $B$ formally smooth over $A$?). ...
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58 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
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17 views

book suggestions on homogeneuos coordinate and projective geometry [on hold]

Will any one introduce some good books about projective geometry and homogeneuos coordinates to me?
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1answer
26 views

Smooth section of Hodge bundle ($F^pH^k$) can be viewed as a smooth form of type$F^pH^k(X,C)$ over$ X$,$ X--->B$ is an analytic family.

I think it is due to Kodaira. could someone explain the idea that Kodaira come up with this. maybe I shouldn't say"can be viewed as". I really mean the smooth form restrict on each fibre is just the ...
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32 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
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30 views

Is Tu's “Introduction To Manifolds” a good place to pick up diff-geo intuition for Vakil's notes?

So I want to study algebraic geometry from Ravi Vakil's notes. However, the only thing I seem to be missing -- I have all the official prerequisites like commutative algebra and point-set topology ...
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22 views

Using Magma to solve a multivariate polynomial system with parameters

I want to solve a system of multivariate polynomials with parameters. Mathematically, the ground field is F = Q(a, b, c, …), the field of rational functions. The polynomials are in F[x,y,z,…]. I ...
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25 views

In the triangle ABC, if BC = [2 [(AC)^2-(AB)^2]/[-AC + sqrt[(AC)^2+4 (AB)^2], prove that 3m(<C) = 2m(<B) [on hold]

Given a triangle ABC, if $$a = \dfrac{2(b^2 - c^2)}{-b + \sqrt{b^2 + 4c^2}}$$, prove that $3m(C) = 2m(B)$.
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39 views

Short exact sequence of groups schemes and dimensions

Let $G$ be a projective groups scheme over an algebraically closed field of positive characteristic $p$. Denote by $G_t$ the $p$-torsion part of $G$ i.e., elements $g \in G$ such that $g^p=0$. Is ...
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3answers
71 views

Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number.

I'm solving some exercises from my class notes on Commutative Algebra,On the following exercise I got stuck: Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number. As far as I ...
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0answers
20 views

For a subspace $S $ in higher dimensions, what is $|S| $?

Let $S\subset \Bbb {A}(k) $. What is $|S|$? I came across this notation whilst studying Algebraic Geometry (conditions imposed by $S $ on polynomials of degree $\leq d $). Thanks!
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93 views

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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24 views

Use of discriminent in proving that the points of unramification is open…

I am confused about Shaferevich Varieties in Projective Space proposition 2.29: If $f : X \to Y$ is a finite map between irreducible varieties, with $Y$ normal, then the set of points in $Y$ over ...
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1answer
35 views

example of computing ramification index

I am trying to understand example 2.2.9 of Silverman's "Arithmetic of Elliptic Curves". In this example, Silverman considers a map $$ \phi:\mathbb{P}^1\to \mathbb{P}^1; [X,Y]\mapsto [X^3(X-Y)^2, ...
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38 views

Is $\mathbb{G}_{m,k}$ (the multiplicative group) simply connected?

I have a field $k$ (which I can take to be algebraically closed if it makes the answer simpler) with the char $k = 0$. The multiplicative group $\mathbb{G}_{m,k}$ is $spec (k [x, x^{-1}])$. ...
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64 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
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3answers
67 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
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25 views

Tangent to dual curve.

Let $C$ be a smooth projective plane curve, let $P \in C(k)$, and let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$, in the dual plane $(\mathbb{P}^2)^*$ (the ...
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2answers
66 views

Show $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. $ k$ field.

Let k be a field. How could I show that $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. I understand that there's a whole algorithm I could go through with Grobner basis, elimination theorem etc. but ...
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1answer
45 views

When is the geometric Picard group $Pic(X_{\overline{K}})$ of finite type?

Let $X$ be a smooth proper geometrically connected variety over a field $K$ of characteristic 0. Let $\overline{K}$ denote an algebraic closure of $K$. What other conditions on $X$ are needed so ...
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45 views

Tangent space of an algebraic variety $ X $ in a point $ x \in X $.

Let $X$ be an algebraic variety, and $ \mathcal{O}_{X,x} $ its local ring at a point $x \in X$, and $ \mathfrak{m}_x $ its maximal ideal. Let set $ k_x = \mathcal{O}_{X,x} / \mathfrak{m}_x $ the ...
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38 views

Is a local equation for a smooth point on a curve given by the equation for the “tangent line”?

Let $X$ be an algebraic curve in $A^m$ defined by some equations $F = (f_1, \ldots, f_n)$. If $p$ on $X$ is a smooth point, general nonsense guarantees that there is a local equation for $p$. Is this ...
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0answers
25 views

Degree of vector bundle under pushforward while blowing up

Let $f:X\longrightarrow Y$ be a birational morphism of projective varieties over $\mathbb{C}$. In particular we can assume that $X$ is a blow of $Y$ at finitely many points. Let $F$ be a vector bundle ...
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2answers
24 views

Parametrization of a sphere

I am trying to argue geometrically that mapping the point $(u,v,0)$ to $(x,y,z)$ gives a parametrization of the sphere $x^2+y^2+z^2=1$ minus the north pole. My questions are: a) What exactly is a ...
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0answers
35 views

Barring a morphism to subvarieties

This is exercise I.3.10 from Hartshorne.I understand that restrict a morphism is continuous but not understand the topological structure of a locally closed irreducible in connection with regular ...
2
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1answer
58 views

Proof of Chow's lemma in EGAII

Section 5.6 of EGAII is dedicated to Chow's lemma. I am having a hard time following an early step of the proof. The version of Chow's lemma in the text assumes that $X$ is a separated $S$-scheme of ...