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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Local ring at a non-singular point and closed subschemes

Let $X$ be a noetherian $K$-scheme, and consider a non-singular point $x\in X$, then a theorem similar to the following should be true : The points of $\text{Spec}\mathcal\, \mathcal O_{X,x}$ are ...
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1answer
14 views

Does every variety contain a smooth subvariety?

Let $X$ be a projective variety (not necessarily irreducible) in $\mathbb{P}^n_{\mathbb{Q}}$. Given any such $X$, does it always contain a smooth subvariety $Y$ and can we say anything about $\dim Y$? ...
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16 views

Algebraic subgroup of ${\mathbf G}_{\mathbf a}^n$

A $1$-dimensional connected algebraic subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$ if the ground field is perfect. Are there any references concerning the ...
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0answers
90 views

I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
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2answers
53 views

What are the prerequisites to understand Algebraic and Differential Geometry?

I have a fair enough knowledge on Real Analysis, Complex Analysis and I am presently pursuing a course in Elementary Abstract Algebra. I actually want to understand the concepts of Elliptic curves and ...
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54 views

What are some topics of advanced number theory every young geometers should know? (soft question)

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some ...
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41 views

Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
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0answers
21 views

Pushforward and pullback of invertible sheaf on ruled surface

Suppose $\pi:X\to C$ is a geometrically ruled surface, and $D$ a divisor on $X$. Then if $D.f=0$ for a fibre $f$, we know by Grauert's theorem that $\pi_{*}(\mathscr{L}(D))$ is a invertible sheaf on ...
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0answers
46 views

Why is the image of an algebraic group by a morphism also an algebraic group?

Let $K$ be a field and $G\subset K^m$ an (affine) algebraic group. If $\varphi:G\rightarrow (K^n,+)$ is a morphism of algebraic groups, why is $\varphi(G)$ is an algebraic group ? I would say for ...
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3answers
94 views

Finding $a^5 + b^5 + c^5$

Suppose we have numbers $a,b,c$ which satisfy the equations $$a+b+c=3,$$ $$a^2+b^2+c^2=5,$$ $$a^3+b^3+c^3=7.$$ How can I find $a^5 + b^5 + c^5$? I assumed we are working in ...
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0answers
19 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
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0answers
17 views

Intersections of connected components of real curves

Let $C_1,C_2\subset \mathbb{P}^2(\mathbb{R})$ be real algebraic curves each of degree $d$. By Bezout's Theorem these curves have at most $d^2$ points of intercestion. Since we are in the real case, ...
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0answers
33 views

Variety of the ideal.

Hey I am trying to understand the inclusion $Z(I) \setminus Z(J) \subset Z(I:J)$through the standard definition of a variety (not the closure). I will be borrowing results from this answer. $$Z(I) = ...
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1answer
9 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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1answer
26 views

Why is a discrete algebraic subset of $K^n$ finite?

Let $K$ be any field. If $A$ is the zero set of a polynomial $P\in K[X]$, then $A$ is finite. This follows from the fact that $K[X]$ is Euclidian, using commutativity of $K$. Now let $A\subset K^n$ ...
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1answer
25 views

Affine varieties and their ideals (part2)

On wikipedia, they talk about varieties $V,W$ and the $I(V)$ and $I(W)$ as well as the quotient ideal, $$I(V):I(W) = I(V - W)$$ Can someone show me a quick proof of the identity?
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23 views

Linear Algebraic Groups with Same Lie Algebra (Soft Question)

Let $G$ and $H$ be two linear algebraic groups over an algebraically closed field $F$ (char 0 ) such that their lie algebras are isomorphic. Now what can we say about the relation between these two ...
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26 views

Simply connected linear algebraic group

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
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1answer
23 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$ [on hold]

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
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27 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
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1answer
69 views

Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
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1answer
29 views

Torsion coherent sheaf on a curve has finite support

I would like to show that a torsion, coherent sheaf $\mathcal{F}$ on a regular integral curve $C$ is supported at a finite number of closed points. This is from Ravi Vakil's notes, namely part 13.7.G. ...
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1answer
90 views

Topology of the complex curve $x^4+y^4=1$

How do you realize that the complex curve $x^4+y^4=1$ looks topologically like three tori glued together with four points at infinity?
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24 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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1answer
20 views

Question on singularity of variety $X$ being irrelevant of choice of polynomials defining $X$

I am getting quite confused with the following material and I would greatly appreciate if someone could provide me an explanation for this. Suppose I have $F_1, ..., F_r \in \mathbb{Q}[x_0, .., x_n]$ ...
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2answers
161 views

Are existentially defined subsets of affine algebraic sets unions of a finite number of affine algebraic sets?

Consider a set of polynomials in $\mathbb{C}[x_1,\dots,x_n]$. The zero locus of these polynomials $Z$ is a subset of $\mathbf{A}^n$ and is an affine algebraic set. Now, consider the following subset ...
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39 views

Monomorphism in the category of schemes

Let $(f, f^{\#}): X \rightarrow Y$ be a map of schemes. The stacks project gives a criterion for $f$ to be a monomorphism (see lemma 25.23.6): if (a): $f$ is a monomorphism in the category of ...
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3answers
61 views

A line avoiding an Algebraic group

Let $K$ be an algebraically closed field, and $G\subset (K,+)^3$ an algebraic subgroup (i.e. given as the zero sets of finitely many polynomial equations) of dimension 1. Is it clear that there is a ...
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0answers
41 views

How to show for a f.g. graded ring $R$, $R^{(m)}$ is generated by degree $1$ for some $m$?

Let $$R=\oplus_{i\geq 0} R_i$$ be a graded ring, which is finitely generated as a $R_0$ algebra. Let $R^{(m)}$ be $\oplus_{i\geq 0} R_{mi}$. Then how to show that for some $m \in \mathbb{N}$, ...
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1answer
30 views

Proving a projective quadric is nonsingular

Let $K$ be an algebraically closed field of characteristic $\neq 2$. Let $C$ be an irreducible quadric curve in $\mathbb{P}^2$, i.e. $C = Z(F)$ where $F$ is an irreducible degree 2 form. I think we ...
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1answer
17 views

Why does the commutator subgroup of a unipotent algebraic group have smaller dimension?

Suppose $U$ is a unipotent linear algebraic group. Is there an explanation why the commutator subgroup $[U,U]$ has strictly smaller dimension, or at least why it is a proper subgroup? This fact is ...
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1answer
52 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
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30 views

Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar cubic graphs, so they are $4$-regular. The ...
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0answers
41 views

Unramified morphism

I was reading the following page: https://ayoucis.wordpress.com/2014/04/06/unramified-morphisms/ and there are several things I do not understand and would like to clarify. First doubt The ...
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1answer
31 views

Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?
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1answer
31 views

Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
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15 views

The induced map $\phi: \mathbb{C}^n \to \mathbb{C}^m$ in the construction of toric varieties

Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous ...
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1answer
37 views

Principal open sets of affine schemes

This question is a special case of Open subschemes of affine schemes are affine? where it is established that in general, open subschemes of affine schemes are not affine. I was wondering if this was ...
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0answers
37 views

The projective space is not affine (II)

This question is closely related to Projective space is not affine. I want to show that the projective space is not affine and to this end I want to prove that $\Gamma(\mathbb P^n_R, \mathcal ...
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0answers
20 views

Higher direct images along the blowup

Let $S$ be a smooth projective surface and $p:X\to S\times S$ be a blowup along the diagonal with the exceptional divisor $E$. How to compute $Rp_*\mathcal{O}_X(-2E)$?
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50 views
+50

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
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59 views

On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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1answer
45 views

What is a divisor (of an algebraic curve)?

So if I have a polynomial $p(x,y)$ and define a curve $C$ based on $p$, what is a divisor? In the context I'm looking at (where I'm trying to learn about Goppa codes), in Joyner et al.'s "Applied ...
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32 views

Lie algebra of a connected simple algebraic group is simple and a simply connected algebraic group having the same Lie algebra

Let $G$ be a connected simple algebraic group over an algebraically closed field $C$. What I infer from this definition is that the defining polynomials of $G$ have coefficients in $C$ while $G$ may ...
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1answer
36 views

Is $X(k')$ a subset of $X$?

Let $X$ be an algebraic $k$-scheme in the sense of these notes (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), and let $k'$ be a field containing $k$. Let $(Y, \mathcal O_Y)$ be the $k$-scheme ...
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1answer
25 views

How to view an inclusion of $k'$-rational points

Let $X$ be an algebraic $k$-scheme in the sense of these notes (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), and let $k'$ be a field containing $k$. By $X(k')$, we mean the set of morphisms of ...
2
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1answer
29 views

Torsion in cotangent sheaf of the cusp

Let $X$ be the cuspidal curve $y^2 = x^3$. This is a singular curve with a unique singularity in the origin. One can easily see this using the Jacobi criterion. How does one show explicitly that the ...
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1answer
48 views

Injectivity of associate map of affine scheme homomorphism

Let $R$ be a ring and the corresponding $(\text{Spec } R, \mathcal{O}_{\text{Spec } R})$ be the affine scheme where $\mathcal{O}_{\text{Spec } R}$ is the structured sheaf of rings. By definition, the ...
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1answer
25 views

Definition of schematically dense

In these notes on algebraic groups (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), assume $(X, \mathcal O_X)$ is an algebraic $k$-scheme for some field $k$, and $S$ is a subset of $X(k)$, where ...
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15 views

Smooth points of the secant variety with a given tangent space

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...