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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43 views

Algebraic solutions to a set of data points

I couldn't find much on this anywhere (probably because I don't know where to look :P).I am wondering, given two points on a plane, is it true that an infinite number of functions of the form: $ax^b + ...
2
votes
1answer
62 views

Kähler differentials of tensor product

Let $B,C$ be $A$-algebras. How can I show that $$ \Omega_{B \otimes_A C/A}=\Omega_{B/A}\otimes_A C \oplus \Omega_{C/A}\otimes_A B?$$
3
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2answers
196 views

Why cubic surfaces contain straight lines?

I often heard that each smooth cubic surface contains even $27$ straight lines. I cannot prove it today, but I'll do my best to do it soon. However how to prove that each cubic surface contains a ...
6
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0answers
50 views

projective varieties and locally trivial fibrations

Suppose $X,Y$ are varieties, $Y$ is projective and $f: X \to Y$ is a locally trivial fibration with fibre $\mathbb{P}^1$. Then there exists an open covering $\{U_i\}_i$ of $Y$ such that $f^{-1}(U_i) ...
0
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1answer
105 views

In the Proj construction, are the homogeneuos prime ideals determined up to an element in the base ring?

Consider the point $[a_0,...,a_n]\sim [a_0\lambda,...,a_n\lambda] \in \Bbb P^k_n$. How do you write the corresponding homogeneous prime ideal in the graded ring $S:=k[x_0,...,x_n]$? Well, the ...
4
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1answer
244 views

SGA 4.5 proof of Hilbert 90 and semilinear Galois action

In SGA 4.5's proof of Hilbert 90, proposition 1.5.2(that the inclusion $V'^G \otimes_k k' \rightarrow V'$ is an isomorphism) is deduced from faithfully flat descent as stated in 1.4.5. The way that I ...
3
votes
1answer
67 views

If $X$ is affine reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$

If $\operatorname {Spec}A$ is reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$ Attempt at a solution: Clearly $D(f) \subset \operatorname{Supp} f$. Since the ...
2
votes
1answer
268 views

What fails in the Cartier <-> Weil divisor correspondence in the singular case?

I know that the following holds in much more generality, but lets say everything happens in the toric case over $\mathbb{C}$. Setting: Given a smooth variety $X$ then there is an isomorphism between ...
5
votes
1answer
90 views

Finding all morphism from a variety to itself

Let $$C:=X^2+Y^2-Z^2$$ be a projective variety in $\Bbb P^2$. What are all the morphisms $C\to C$ ? More generally, how does one find all morphisms from a given variety to itself? ...
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48 views

Show that $\operatorname{Spec}k[x_1,x_2,…,x_n]/(x_1^2+\cdots+x_m^2)$ is normal for $\operatorname{char}k\neq 2, n\ge m \ge 3$ [duplicate]

I want to show that if $F(T) \in B[T]$, where $B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)$, is monic and has a root $\alpha \in\mathcal K(B)$ then $\alpha$ actually lives in $B$. This will imply that ...
3
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2answers
109 views

$Im(\phi)$ is closed subset of $\mathbb{A}^2$

let $\alpha(t)$ and $\beta(t)$ $\in$ $K[t]$ , $\phi(t)=(\alpha(t),\beta(t))$ is a morphism from $\mathbb{A}^1$ to $\mathbb{A}^2$ show that $Im(\phi)$ is closed subset of $\mathbb{A}^2$. it seems ...
2
votes
1answer
63 views

Working out $\operatorname{Proj} k[x_0,…x_n]/(x_0^2,x_0x_1)$

Let $I$ be the homogeneous ideal given by $(x_0^2,x_0x_1)$ and describe $X= \operatorname{Proj} k[x_0,...x_n]/I$. My goal here is to see that even though $I$ is not a radical ideal, X is still a ...
1
vote
1answer
129 views

On the Grothendieck ring of varieties

The Grothendieck group of varieties $K_0(\textrm{Var}_k)$ over a field $k$ is the Abelian group generated by isomorphism classes of quasi-projective $k$-varieties, subject to the scissor relation ...
4
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1answer
187 views

Identifying the stalk of the general point of an integral scheme with the field of fractions of any open affine

Let $X$ be an integral scheme and $\eta$ its general point. Then we can identify $\mathcal O_{X,\eta}$ with $FF(A)$ where $\operatorname{Spec} A$ is any open affine of $X$, because $\eta$ lives in all ...
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2answers
95 views

In a quasicompact scheme every nonreduced point has a closed nonreduced point in its closure

Let $x\in X$ be a non reduced point. Then I can find a nilpotent $f\in \Gamma(U, \mathcal O_X)$, where $U$ is some open neighborhood of $x$. I also know that when $X$ is assumed to be quasicompact, ...
0
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1answer
40 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
2
votes
0answers
70 views

The dimension of a projected variety

Suppose I have an algebraic variety $V\subset \mathbb{R}^5$, with dimension 2 and degree d. Consider the projection $V_0=\{x_1,x_2,x_3|x_1,...,x_5 \in V\}$. While $V_0$ might not be an algebraic ...
6
votes
2answers
69 views

Some notes, literature on grassmanians

Can anybody provide a link to some notes on grassmanians? I mean something 'elementary': description of Plucker embedding, lines and hyperplanes on grassmanians and so on.
4
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1answer
139 views

Book with color pictures of algebraic surfaces

I have a pretty specific question: I'm looking for a book with color pictures of algebraic surfaces. Could anyone point me in the right direction?
9
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1answer
330 views

How do people define flop?

Let $f:Y \to X$ be a small contraction morphism of projective normal varieties. Then what is the usual way (if any) to define the flop $f^+: Y^+ \to X$ ? I could find a clear and uniform way to ...
2
votes
1answer
87 views

Example where prime spectrum suits better than the maximal spectrum

in a lot of algebraic geometry books I've heard that working over $\mathrm{Spec}(A)$ is better than working over $\mathrm{Spmax}(A)$ in the case where you consider a variety over a non-agebraically ...
4
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1answer
48 views

Integer solutions to an ellipsoid surface

Given the equation $$x^2+2y^2+5z^2+xz =n$$ where $n$ is any positive integer, what is the smallest odd integer for which no integer solution $(x,y,z)$ exists (i.e. $x,y,z$ are integers)? I know that ...
6
votes
1answer
100 views

Showing that an algebraic set is not isomorphic to $\mathbb{A}^1$

(For convenience, I'm assuming $\mathbb{K} = \mathbb {C}$.) I'm trying to show that the algebraic set $\mathbb{V}(xz-y^2,x^3-yz,z^2-x^2y) \subseteq \mathbb{A}^3$ is not isomorphic to ...
5
votes
2answers
152 views

What is the coordinate ring of $G/U$?

Let $G$ be an algebraic group and $U$ its subgroup consisting all upper triangular matrices. For example, $G=GL_n(k)$ and $U$ the subgroup consisting of all upper triangular unipotent matrices in ...
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0answers
145 views

$V$-regular lines, finite morphisms

Let $V$ be a linear space over $k$ of dimension $n$. Consider an affine algebraic variety $X$ and $x$ a finite morphism of algebraic varieties $v : X \to V$. We say that a line $l \subset V$ (i.e. a ...
5
votes
0answers
173 views

Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto ...
3
votes
0answers
64 views

Valuative criteria with varieties

Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over ...
2
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140 views

Criteria for a contravariant functor from Schemes to Sets to be representable by a scheme

I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's ...
8
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1answer
172 views

Linear combination of matrices

Let $A, B, C, D$ be four linearly independent symmetric 3 x 3-matrices over $\mathbb K$. Show that some linear combination of these matrices is a matrix of rank 1. I know it is supposed to be a ...
2
votes
0answers
51 views

The two possible structures on a triple point

Let $k$ be an algebraically closed field. I would like to prove that there are only two possible $k$-scheme structures on a triple point, namely that of a $2$-jet $\textrm{Spec }k[x]/(x^3)$, and that ...
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0answers
193 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
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55 views

Number of components of algebraic variety

For (real or complex) algebraic varieties given by some explicit polynomials, are there methods to compute the number of connected components?
2
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1answer
150 views

Is the image of this regular map open, closed?

This is a question from Shafarevich that I'm trying to figure out. It asks if the image of $f(x,y,z)=(x,xy,xyz)$ is open or closed (in relation to the Zariski Topology)...I think it is neither but I'm ...
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votes
1answer
85 views

Flatness question

In reading on the stacks project I came across a result I don't quite follow: "Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,…,xr∈M ...
3
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0answers
204 views

Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
3
votes
1answer
82 views

Question on Nakayama?

In reading a certain proof on the stacks project "http://stacks.math.columbia.edu/tag/00NV", I can't see how Nakayama's lemma is used to make the following conclusion: "Assume M is finitely presented ...
0
votes
1answer
97 views

Symmetries of a square and it's similarity to the Division Algorithm

I need help with this question: (each variable $r$ represents a rotation of the square about the axis through its centroid at $90^{\circ}$ intervals. $e$ represents nonmotion. This question is ...
2
votes
0answers
156 views

Castelnuovo's rationality criterion following Beauville

Following Beauville's book "Complex algebraic Surfaces", in order to prove Castelnuovo's rationality criterion i need to prove one lemma and one proposition. There is one point of proof of lemma V.8 ...
7
votes
1answer
163 views

Difference between graded ring and graded algebra

Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring. Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...
4
votes
1answer
269 views

Hyperplane sections on projective surfaces

I am studying Beauville's book "Complex Algebraic Surfaces". At page 2 he defines the intersection form (.) on the Picard group of a surface. For $L, L^\prime \in Pic(S)$ ...
2
votes
1answer
232 views

What is the group of $k$-rational points of an algebraic group?

Let $k$ be a field and $G$ a linear algebraic group over $k$. What is the group of $k$-rational points of $G$? By definition, $G$ is an algebraic variety. Suppose that $G$ is defined by polynomials ...
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0answers
30 views

Reduction of isogenies at bad primes

Let $L$ be a number field and $E,E'$ two elliptic curves defined over $L$. Suppose $\varphi\colon E\to E'$ is an isogeny defined over $L$. Let $\mathfrak p$ be a prime of bad reduction for $E,E'$. ...
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2answers
154 views

Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$

I am giving a simple introductory course on algebraic geometry and I plan to mention that $$\mathbb RP ^3\cong SO(3,\mathbb R).$$ I know a rather simple proof of this using the fact that $\mathbb ...
5
votes
2answers
160 views

On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ...
5
votes
2answers
208 views

Function field of an affine hypersurface

I am reading Hulek' Elementary Algebraic Geometry, p.103 Let $V$ be an irreducible affine hypersurface, say $V=V(f)\subset\Bbb{A}^n$. Then the coordinate ring is by definition ...
2
votes
2answers
273 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
3
votes
2answers
175 views

Parametrization of the line in the projective space

Let $L=aX+bY+cZ$ be a line in the projective space, the book I'm using states that every such line has the following parametrization: $$\varphi:\mathbb P^1\to L, \ (t:s)\mapsto ...
4
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0answers
224 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
1
vote
2answers
54 views

Calculating a circle's radius from one point and two circles on it's circumference

Suppose that there are four points $A, B, C, D$. A circle of radius $r_A$ surrounds point $A$, a circle of radius $r_C$ surrounds point $C$, and a circle of radius $|DB|$ surrounds point $D$. ...
1
vote
1answer
40 views

Ring of rational functions ideal generators

There is an affine variety $X\subset \mathbb{A}^n$ with its ring of rational functions which is the quotient ring of $\mathbb{k}[X]$ (each $f\in \mathbb{k}(X)$ has a form $\frac{p}{q}$ where $q$ does ...