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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11 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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1answer
23 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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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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0answers
134 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 derivations of the algebra $R$ endowed with the lie ...
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2answers
146 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ...
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29 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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0answers
19 views

A computation of a dimension of a space of sections

Let $S$ be a finite set with $m$ éléments 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
26 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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38 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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1answer
41 views

What is the order of a cusp form at a cusp?

This question is about the definition of order of a section of a bundle at a point, and the related notion of associated divisor. Let us look at a specific example, the discriminant $\Delta(z)$ on ...
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33 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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17 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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1answer
35 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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1answer
21 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 ...
2
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0answers
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
33 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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0answers
26 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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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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0answers
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
337 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
5
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1answer
77 views

Proof of Serre duality for $D=0$

I have been working through a proof of Serre duality, which proceeds by induction on the divisor $D$, but I am having trouble with the base-case. How can I prove that on a riemann surface X, $H^0(X, ...
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206 views
+100

Weighted projective space and $\mathrm{Proj}$

I'm trying to solve a problem from Jenia Tevelev's notes on GIT. (Problem 5 at the end of this pdf.) Compute $$\operatorname{Proj}\frac{\mathbb{C}[x,y,z]}{(x^5+y^3+z^2)}$$ where ...
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24 views

When are the integral cohomology groups of a toric variety free?

Let $X=X_{\Sigma}$ be a complete, toric variety associated to the fan $\Sigma$. I am interested in conditions on $\Sigma$ (or on the polytope $P$ when $\Sigma$ is the normal fan of $P$) that guarantee ...
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52 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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0answers
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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2answers
69 views

Proof that $H^{0,1} \oplus H^{1,0} = H_{DR}^1$

I am struggling with a proof from Donaldson's Riemann Surfaces which he leaves as an exercise. we want to construct an isomorphism from the direct sum of $H^{1,0}(X)$, the set of holomorphic 1-forms ...
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0answers
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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0answers
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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0answers
21 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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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
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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0answers
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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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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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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0answers
91 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.
3
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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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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 $spec (k [x, x^{-1}])$. ...
2
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0answers
70 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
3
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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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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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0answers
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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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
44 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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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 ...
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 ...
6
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0answers
91 views

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
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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 ...
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2answers
29 views

Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in ...