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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How to Distinguish Between Base-points in Blowups?

As an example consider the resolution of the base-point (via blowups) of the family of curves in $\mathbb{C}^2$ defined by $f(x,y)=4x^3-ax-b-y^2=0$, where $a$ is a fixed constant and $b$ is a free ...
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1answer
42 views

Variety that is affine and projective is a finite number of points

I was trying to proof the following without any luck. I would appreciate good hints. A projective variety that is isomorphic to an affine variety is a finite number of points.
2
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1answer
39 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
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2answers
100 views

How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook. The skillset I am least satisfied with is the application of the Category Theory to Algebraic ...
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1answer
34 views

Ideals agreeing in a localization

I have an integral scheme $X$, and two coherent ideal sheaves $\mathcal I$ and $\mathcal J$ on $X$. I know there is a (maybe not closed) point $x$ of $X$ such that $\mathcal I$ and $\mathcal J$ ...
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2answers
50 views

Curves with negative self-intersection

Let $S$ be a non-singular projective surface over $\mathbb{C}$. Show that $S$ contains at most countably many irreducible curves $C$ with $C^2<0$.
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1answer
33 views

Angles of diagonals in a quadrilateral

I have a quadrilateral with known angles. Also known is that edge BC and CD have the same length. How can I find out the ratio the diagonals divides the angle α into α1 and α2?
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1answer
29 views

Infinite Varieties and Non constant Common Factors

I'm trying to work out some problems from Ideals, Varieties, and Algorithms, and I've stumbled on one that I'm unsure of how to start: Let $f,g \in \mathbb{C}[x,y]$ be nonzero. In this exercise, ...
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1answer
32 views

Rational points of projective spaces over rings

Let $X=\mathbf{P}^n_A = \text{Proj} A[T_0,\ldots,T_n]$. If $A$ is a field, there is a simple classical description of $X(A)$. However, if $A$ is a more general ring, like $\mathbf{Z}$, I don't see an ...
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1answer
27 views

Can a map from a p-simplex to the n-sphere be surjective?

If $p<n$, is this possible? I am confused about this. I am trying to prove that the i-th singular homotopy group of the n-sphere is a subset of the i-th homotopy group of $\mathbb{R}^n$ but I am ...
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25 views

Are generalised configuration spaces related to holomorphic maps?

A branched cover of the Riemann sphere is a non-constant holomorphic map $\phi: \Sigma \to \mathbb{C}P^1$ where $\Sigma$ is a compact Riemann surface. The Hurwitz space of branched coverings of the ...
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1answer
49 views

The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local ...
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1answer
41 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
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49 views

Flatness and Cohen-Macaulay rings

Let $A$ be a local Artin ring, $R$ a local Noetherian ring, $f:A \to R$ a flat morphism and $R$ is cohen-Macaulay. Let $I$ be an ideal in $R$ such that $R/I$ is also Cohen-Macaulay. Under what ...
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1answer
53 views

Definition of $\mathbb{A}^n_S$ by glueing

In Eisenbud and Haris (Geometry of schemes I.2.4), if $S=\cup_\alpha U_\alpha$ with the $U_\alpha$ affines, to define $\mathbb{A}^n_S$ one take $X=\cup_\alpha \mathbb{A}^n_{U_\alpha}$ with ...
4
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2answers
243 views

What do we lose in Projective Spaces?

We can think of the Complex Numbers as an extension of the Real Numbers, similarly we can think of the Projective Plane naturally as a nice extension of the Euclidean Plane. But, when we go from real ...
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0answers
19 views

Finding equations of locus given its directed distances

A point moves so that the product of its directed distances from the lines 3X+4Y-7=0 and 3X-4Y+1=0 is 144/25. Find the equations of its locus. What curve is it?
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1answer
55 views

Proving $V(f)$ is finite when $f$ is non-constant

My problem asks me to show that if $f$ is non-constant, then $\mathbf{V}(f)$ is finite. Assume that $f \in \mathbb{C}[x]$. If $f$ was an ideal, this would be straightforward; however, $f$ is merely ...
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1answer
27 views

On the isomorphisms $(\mathcal{O}_{Z,X})_\mathfrak{p}\cong\mathcal{O}_{Y,X}\text{ and }\mathcal{O}_{Z,X}/\mathfrak{p}\cong\mathcal{O}_{Z,Y}$.

Suppose you have two closed, irreducible subvarieties $Z\subseteq Y$ in some variety $X$. (I'm not sure if it matters, but for ease I'll just assume everything is over an algebraically closed field.) ...
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2answers
43 views

quotient is a projective variety

Studying some introductory algebraic geometry (affine and projective varieties) I came up with this which I can't understand: $K$ is an algebraically closed field. If we define a map $ f : SL_2 ...
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1answer
31 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
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2answers
50 views

Seamlessly connect a sine curve and a parabola

I want to seamlessly connect an unknown parabola to a known sine wave. The equations are: s(x) = a sin(bx + c) p(x) = Ax^2 + Bx + C I want to draw ...
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276 views
+50

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
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2answers
61 views

Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?

Let $R$ be a ring. $p$ be a prime ideal of $R$ . Let $R\rightarrow R_p$ be the canonical. Consider the map of schemes $Spec(R_p) \rightarrow Spec(R)$. Is it an open immersion. $Spec(R_p)$ is an open ...
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1answer
62 views

A point in $ PGL(R) $ not in $ GL(R)/R^{\times} $

A bit of notational background first. Let $k$ be a field and define $ PGL_{n} = Spec(k[x_{ij}]_{(det)}) $, where $i,j = 1,...,n$ and where $k[x_{ij}]_{(det)}$ denote the degree $0$ part of the graded ...
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27 views

quadratic constraints representations

I have two surfaces of parameter $x \in \mathcal{X} \subset \mathbb{R}^n$ with $\mathcal{X}$ to be a polytope: $u(x) = ax + b \subset \mathbb{R}^m, z(x) = x^TAx + B^Tx + C \in \mathbb{R}$. If $m=1$, ...
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0answers
47 views

An elementary Algebraic Geometry text, similar to Kempf's Algebraic Varieties

Is anyone familiar with an elementary Algebraic Geometry book, which takes a similar approach to that of Kempf's Algebraic Varieties, but is more user friendly ?
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0answers
41 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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17 views

Criterion for $\mathcal{O}_{Y,X}\cong\mathcal{O}_{Y',X'}$ for $Y,Y'$ closed, irreducible subvarieties.

Suppose you're working over an algebraically closed field $F$, and let $X$ and $X'$ be quasi-affine varieties, with $Y\subseteq X$ and $Y\subseteq X'$ closed, irreducible subvarieties. I read the ...
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0answers
39 views

Showing a divisor is effective

On $\mathbb{P}^n$, consider the prime divisor $C=Z(x_0)$. Let $f \in k[D(x_0)]$ be degree $d$. I want to show that the divisor $(f)+\nu_C(f)C$ is effective. Now, $f=g/x_0^d$ with $g \in ...
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1answer
22 views

A question about a universal family over a Grassmannian.

I refer to this paper on Moduli Spaces by Ravi Vakil. I am uploading a screenshot: What can possibly be a universal family over $G(k,n)$? For example, let us take the set of all linear subspaces ...
2
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1answer
29 views

Dominant Rational Map

Let $Y\subseteq \mathbb{A}_k^n$ be an affine variety and $X$ any other (quasi)-(projective) variety with $U$ an open-subset of $X$. If $\theta: A(Y) \to \mathcal{O}(U)$ is a $k$-algebra homomorphism ...
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1answer
24 views

Finding a valuation

Shafarevich, in Basic Algebraic Geometry I, makes the following definition, which I'm having trouble understanding in a concrete example. Let $X$ be a variety, nonsingular in codimension 1, and let ...
2
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2answers
54 views

Two Projective closures of $\mathbb{A}^1\setminus\{0\}$ which are not isomorphic as varieties.

I am doing some sample problems for my upcoming Algebraic Geometry exam, and one of the questions is: Is it true that all projective closures of an affine variety $X$ are isomorphic as ...
5
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1answer
83 views

Castelnuovo-Mumford regularity of a Veronese subring

I've faced a problem while reading a paper. It is mentioned to be trivial but I couldn't prove it. I'd appreciate if you can lead me to some resources or if you can prove it for me. Thank you. ...
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1answer
16 views

On the wording of a question related to open cover of sets

I am working on the exercise where the hypothesis is : Let $X$ be a scheme such that there exists affine open subsets $U_i \ (1 \leq i \leq n)$ such that $X = \cup U_i$. Further any two of the $U_i$'s ...
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2answers
99 views
+50

The intersection points are collinear

It is given a hexagon inscribed in a conic section. I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear. How could we do this? ...
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1answer
145 views

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
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1answer
21 views

Closed subscheme of a projective scheme determined by homogeneous ideals

So in Ravi Vakil's notes Ex 8.2C, I have to prove that if $\pi:X\hookrightarrow\text{Proj}\ S_{\cdot}$ is a closed subscheme (here $S_{\cdot}$ is a graded ring finitely generated by elements of degree ...
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1answer
49 views

What is the Euler characteristic of a determinantal hypersurface?

If $X\subset \mathbb P^{n}$ is a smooth (complex) hypersurface, one can compute its topological Euler characteristic $\chi(X)$ by taking the degree of the $0$-cycle $c_{n-1}(T_X)\cap [X]$. If $X$ is ...
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54 views

Classes of rings C[x,y]/(x²+cy²+ey+f) [duplicate]

I have a question. I would like to describe the classes of rings that appear in $\mathbb{C}[x,y]/I$ up to isomorphism, where $I=(Q)$, $Q=x²+cy²+ey+f$, $c,e,f\in\mathbb{C}$. $Q$ comes from ...
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38 views

A basic question on the algebraic dimension of an algebraic variety

Let $W,V\subset \mathbb{C}^{2n}$ be irreducible varieties both of dimension $n$ defined as the $0$-sets of some prime ideals $I$, $J$ on $\mathbb{Q}[x_1,\dots,x_{2n}]$, ideals which remains prime in ...
3
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1answer
36 views

Is the $n$-sphere $x_1^2+\cdots+x_n^2-1=0$ a rational variety in $\mathbb{A}^n$?

I asked a question a few days ago about where the function field $k(x,\sqrt{1-x^2})$ was purely transcendental over $k$, for $k$ algebraically closed. It turned out to be true, so I know this proves ...
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1answer
47 views

Can arithmetic and geometric genus be arbitrary?

I'm reading about genuses from Liu's book "Algebraic Geometry and Arithmetic Curves". There he defines arithmetic genus of a projective curve and geometric genus of a smooth projective variety. As far ...
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2answers
140 views

Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
3
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1answer
100 views

Describe the topology of Spec$(\mathbb{R}[x])$

I am supposed to describe the points and topology of Spec$(\mathbb{R}[x])$, I managed to describe the points but I dont understand the "topology" of the set, what does this mean? Are they asking for ...
2
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1answer
58 views

Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by this scheme?

Let $S$ be a scheme and $\mathcal {O}_S$ the structure sheaf of rings over $S$. Question: Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by a ...
1
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1answer
32 views

Definition of generic point over finite set

I am currently reading David Marker's paper "A Remark on Zilber's Pseudoexponentiation" (J. Symb. Logic 71 (2006), no. 3). He describes the axioms for Zilber's pseudoexponential fields. The Strong ...
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1answer
79 views

Which categories of linear representations are semisimple?

Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) ...
2
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1answer
37 views

algebraic equivalence of line bundles

Two line bundles $L,L'$ on $X/k$ are called algebraically equivalent if $L = M|_{X\times t}$ and $ L'=M|_{X\times t'} $ for some line bundle $M$ on $X \times T$ with $T$ smooth and irreducible and two ...