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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31 views

How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
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1answer
42 views

On the anti-equivalence of affine schemes with commutative rings

There is an equivalence $\mathbf{Aff}\simeq \mathbf{CRing}^{\text{op}}$ between the category of affine schemes and the category opposite to the category of all commutative rings. If we instead ...
3
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1answer
130 views

Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials

I am reading Gathmann's free online notes on Algebraic Geometry. One exercise asks to show that "Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero ...
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0answers
26 views

Prove that over an infinite field, a finite set of points in $\mathbb{ A}^n$ can be obtained as vanishing set of n polynomials [duplicate]

While reading Ernst Kunz's commutative algebra book, I came across this problem: Let $K$ be an infinite field, and $V \in\mathbb{ A}^n (K)$ (the affine n-space) be a finite set of points. Show ...
4
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1answer
65 views

A proposed criterion for finding when an homogenous ideal is radical

Let $X$ be a projective variety over an algebraically closed field, and $I$ be the homogenous ideal of $X$ and $J$ be an ideal with the same zero set. Suppose that I know $I=\langle f_1,...f_n ...
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2answers
1k views

Software for solving geometry questions

When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every ...
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0answers
16 views

When does the equality $\mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p}=\dim R$ happen? [duplicate]

In the context of Krull dimension, given any commutative ring $R$ and $\mathfrak{p}\subset R$ a prime ideal, we have (almost by definition) $$ \mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p} ...
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0answers
34 views

Intersection of hypersurfaces in the projective space

Fix an integer $n>0$. Is it true that for any $k>0$ and a closed point $x \in \mathbb{P}^n$, there exists hypersurfaces sections of degree $k$ (i.e., global sections of ...
2
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1answer
31 views

How to view a morphism of Lie algebras?

Let $G = \textrm{GL}_n k$, and let $\sigma: G \rightarrow G$ be an automorphism of algebraic groups. The Lie algebra $\mathfrak g$ of $G$ can be described in three ways: 1 . The space $T_e(G)$ ...
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0answers
29 views

Understanding functions in $\mathbb{P}^1$

I just need solution check: I am given function $f\in k(\mathbb{P}^1)$ that sends $(x:y) $ in $\frac{x}{y}$. If I understood this right, this is just coordinate function $x$ since it send and point ...
2
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32 views

Cremona Transformation and isomorfism

Let $P_1=(1:0:0),P_2=(0:1:0),P_3=(0:0:1) \in \mathbb{P}_2$ (over an algebraic closed field). Denote $U=\mathbb{P}^2 \setminus \{P_1,P_2,P_3 \}$ and consider the map $$ f:U \rightarrow \mathbb{P}^2, ...
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3answers
82 views

Motivation for rings of fractions?

I'm learning about rings of fractions and localization. I like the material a lot and feel engaged with it, but I do lack a broader perspective on things. For example, I'm aware of things such as ...
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1answer
18 views

Hypersurfaces have no embedded points (Vakil 5.5.I)

Here's a question from Vakil's FOAG. If $f\in k[x_1,\ldots,x_n]$ is non-zero, show that $A:=k[x_1,\ldots,x_n]/(f)$ has no embedded points. Hint: suppose $\bar{g}\in A$ is a zero-divisor, and choose ...
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1answer
42 views

Twisting an exact sequence of sheaf gives an exact sequence

Reading through Hartshorne's Algebraic Geometry, and reading different notes about cohomology of sheaves, I have often seen the argument that if you have an exact sequence of coherent sheaves over a ...
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0answers
44 views

Finding genus of projective curve

Can anyone help me in finding the genus of the curves a) $x^2y^2-z^2(x^2+y^2)$ b) $(x^3+y^3)z^2+x^3y^2-x^2y^3$ c) $y^4+z^4-2x^2(y-z)^2$ d) $y^2z^2-x^4-Y^4$ e) $(x^2-z^2)^2-2y^3z-3y^2z^2$ Here ...
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19 views

Hyperelliptic Curves and Period Integrals

I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic ...
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0answers
40 views

Relating a fiber of a sheaf and its cohomology, from Huybrechts & Lehn - The Geom. of Moduli Spaces of Sheaves

Reading the proof of lemma 4.4.4 in Huybrechts and Lehn Geometry of Moduli Spaces of sheaves I come across an isomorphism relating a fibre of an invertible sheaf and its cohomology, and I really don't ...
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0answers
29 views

Relation between pullbacks of a degree zero line bundle on an elliptic curve

Let $E$ be an elliptic curve over a field $k$. Let $$\mu:E \times_k E \to E$$ be the addition map on $E$. Furthermore let $p_1,p_2:E \times_k E \to E$ be the two canonical projections and let ...
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0answers
29 views

How to know if a triangle contains any of the dots

suppose in a 2D space, there is a grid of dots/points at every integer x,y (e.g. 1,1 2,1 2,2 etc) . Now you have a triangle with points (a,b) (c,d) and (e,f), which may not be integer. Now I want to ...
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1answer
195 views

Difference between affine variety and graph of a function

I just started to read Algebraic Geometry and when I was going through exercises in Hartshorne's algebraic geometry book I came about the following question. Consider the plane curve $y = x^2$. Then ...
2
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1answer
65 views

$P$ prime implies $V(P)$ irreducible?

Let $P\subset k[x_1, \ldots, x_n]$ be a prime ideal. Is it true that the variety $V(P)$ is irreducible? This is easy to show when $k$ is algebraically closed. Is it also true in general?
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1answer
73 views

Do I understand the Chevalley Restriction Theorem correctly?

Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map ...
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2answers
79 views

Bertini's Theorem and singular divisors on a surface

I'm trying to understand the following: Let $X$ be a projective, smooth surface over an algebraically closed field and $D$ a divisor on $X$. How can I see that $D$ is linear equivalent to the ...
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1answer
26 views

What are the coordinates for the center of the second circle? (Full question in body)

Full Question:A circle has its center at (6,7) and goes through the point (1,4). A second circle is tangent to the first circle at the point (1,4) and has one-fourth the area. What are the coordinates ...
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22 views

Hilbert scheme of quasi-projective variety

Suppose $X$ is a projective scheme over an algebraically closed field $k$, denote its Hilbert scheme with Hilbert polynomial $p$ by $\text{Hilb}^p_X$, then from section 1.1 of Nakajima's book, ...
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2answers
110 views

Betti numbers of complex “sphere”

Let $X$ be the set of solutions to $x_1^2+\ldots+x_n^2=1$ in $\mathbb{C}^n$. This has real dimension $2(n-1)$, but since $X$ is an affine algebraic variety, the only possible non-zero topological ...
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0answers
41 views

Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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0answers
37 views

surfaces in $\mathbb{C}^{2}$ cut out by the equation $y^{2}=x^{3}+ax + b$

I want to show that the solution set of the equation $y^{2}=x^{3}+ax+b$ in $\mathbb{C}^{2}$ is isomorphic to punctured torus. I have made the following effort, first of all i have tried to find the ...
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1answer
28 views

Characterization of the elements of a quotient ring

I'm in trouble with the following exercise: Consider the ideal $ I = (X^2-Y^3,Y^2-Z^3) $ in the polynomial ring $ k[X,Y,Z] $, where $k$ is any algebraically closed field. Show that every element of $ ...
2
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1answer
54 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism ...
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1answer
48 views

Linear algebra prerequisites for abstract algebraic geometry

I'm interested in what linear/multilinear algebra does one need to study algebraic geometry(following EGA and Harthshorne). Texts I have in mind are like "Foundations of algebraic geometry" by Ravi ...
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0answers
16 views

Cohomology of Grassmanian: pairing with fundamental class

Let $Gr(k, V)$ be a Grassmannian with $\dim V=n$, and $S$ be a tautological bundle over $Gr(k, V)$, so $\operatorname{rank} S=k$. Then the cohomology ring $H^*(Gr(k, V))$ is generated by Chern classes ...
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1answer
15 views

Why is conjugation by a diagonal matrix a semisimple automorphism of $\textrm{GL}_n$?

Let $$s = \begin{pmatrix} \lambda_1 & & & 0\\ & \lambda_2 & \\ & & \ddots \\ 0& & & \lambda_n \end{pmatrix}$$ be a diagonal invertible matrix. Let $G = ...
2
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1answer
28 views

Show that this morphism of varieties is not separable

Let $k$ be an algebraically closed field of characteristic $2$, $G = \textrm{SL}_2(k)$, and $z = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$. Let $\sigma: G \rightarrow G$ be the ...
3
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1answer
155 views

Alternative description of the sheafification

For me the sheafification of a given presheaf is this: Proposition-Definition: Given a presheaf $\mathscr{F}$, there is a sheaf $\mathscr{F}^+$ and a morphism $\theta \colon \mathscr{F} \to ...
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0answers
46 views

How to show $\text{Sym}^n(\mathbb{P}^1)=\mathbb{P}^n$

From the question Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): ...
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0answers
17 views

Symmetric Product of a Projective scheme

Following the question, Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): ...
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0answers
12 views

A function vanishing on the subgroup generated by relations defines a linear function.

I am reading Basic Algebraic Geometry 1 by Shafarevich (3rd edition) and I couldn't understand the following portion on pg 222: Namely, in Section 5.2 we defined the module of differentials ...
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1answer
26 views

Solving nonlinear system algebraically

I have the system of equations: $$2x(1+\lambda)=0$$$$2y(1+\lambda)=0$$$$2z(1-\lambda)=0$$$$x^2+y^2-(z^2+1)=0$$ It's easy to plug in a few values and see that the solution is $x^2+y^2=1$, $z=0$, and ...
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0answers
74 views

Why Flat module is 'flat'? [duplicate]

I've been reading about 'flat module' in Dummit and foote algebra book. It is a R-module that preserves exactness after tensoring it to any exact sequence of R-module. (i.e., It preserves ...
6
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1answer
51 views

Vakil's Foundations of Algebraic Geometry, Exercise 7.3.F

$\DeclareMathOperator{\Spec}{Spec}$ I'm having trouble with the exercise in the title, which states "Suppose $Z$ is a closed subset of an affine scheme $\Spec A $ locally cut out by one equation. ...
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0answers
47 views

Infinite primes (places) of a number field geometrically

Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local ...
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24 views

Dimension of the fibre product of varieties

Let $\mu : X \to Y$ be a morphism of algebraic varieties. Suppose that both $X$ and $Y$ are of pure dimension $x$ and $y$, respectively. Under what assumptions can we conclude that the fibre product ...
3
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1answer
32 views

Kummer surfaces are smooth

Let $X$ be the Kummer surface associated to an abelian surface $A$. I will denote by $\epsilon : \tilde{A} \rightarrow A$ the blow-up of $A$ at the 16 fixed points of the involution $i : A \rightarrow ...
4
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1answer
44 views

The morphism defined by a linear system associated to a smooth curve of genus 2 on a $K3$ surface has degree 2 and its branch locus is a sextic.

This is part of Proposition $VIII.13$ in Beauville's "Complex algebraic surfaces": Let $S$ be a $K3$ surface and $C \subset S$ a smooth curve of genus $g=2$. Then the morphism $\phi$ defined by the ...
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22 views

Universal family of the Hilbert scheme of one point.

If $X$ is projective over a base scheme $S$, then its Hilbert scheme of one point is just itself. i.e. \begin{equation} \text{Hilb}^1(X)=X \end{equation} What about the universal family? i.e. there ...
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30 views

Hilbert scheme of $n$ points on a smooth curve

If $C$ is a smooth curve over a field $k$, then from lots of references, e.g. Janos Kollar, Rational Curves on Algebraic Varieties, exercise 1.4.1, that the Hilbert scheme of $n$ points is ...
2
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1answer
33 views

Morphism whose fibers are finite and reduced is unramified

Definition: Let $f: X \to Y$ be a morphism of finite type of locally Noetherian schemes, $x \in X, y = f(x) \in Y$. Say that $f$ is unramified at $x$ if the map on stalks satisfies $m_y ...
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1answer
32 views

Visualizing sections of nontrivial vector bundles

My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the ...
2
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1answer
2k views

Intersection of 2 lines in 2D via General Form Linear Equations

). Recently, I asked you how to find the "Intersection of 2 Lines in 2D" and the answers revolved around the Determinants ( http://en.wikipedia.org/wiki/Line-line_intersection ) or Systems ( ...