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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Classic Circle and Adjacent Arrangement Problem

Given: A circle with nine distinct positive integers. n is a fixed positive integer. So these positive integers are arranged in the circle in such a way that the product of any selected two ...
3
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1answer
34 views

isomorphism of pointed sets

What is an isomorphism in the category of pointed sets? Is it just an exact sequence $$ 1 \to A \to B \to 1 ?$$ (Note: even though the kernel of the middle map is zero, $A$ might not inject into $B$.) ...
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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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1answer
54 views

Book recommendations for topics leading upto Algebraic geometry

I'm interesting in studying algebraic geometry (specifically either from Shafarevich or Hartshorne). Assuming a high school and basic college math education, what should be the topics and the order ...
2
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1answer
49 views

Abstract Varieties

Hartshorne does not seem to bring this concept up so far in his AG book but I am guessing that one may define an "abstract variety", in a similar way as one defines an abstract manifold from DG. ...
3
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1answer
69 views

Hartshorne Exercise III 6.2 (a)

Let $X=\mathbb{P}^1_k$, with $k$ an infinite field. Show there does not exist a projective object $\mathcal{P}\to\mathcal{O}_X\to 0$. The author suggests to consider surjections of the form ...
2
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1answer
35 views

What is the dimension of $A-B$, where $B$ is a subspace of $A$?

My question is really simple, what is the dimension of $A-B$, where $B$ is a subspace of $A$? this space is well-defined? I found this space in this paper on page 440: Following my calculations in ...
6
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2answers
149 views

On the $j_!$ of a sheaf

Let $X$ be a topological space, and $U$ an open subset. Denote $j:U\to X$ the inclusion. Let $\mathcal F$ be a sheaf on $U$. We define $j_!\mathcal F$ to be the sheaf associated to the presheaf ...
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1answer
73 views

Why can schemes of finite type over $\mathrm{Spec}\left(k\right)$ be considered to be affine?

Let $k$ be a field (not necessarily algebraically closed). We call $k$-variety a scheme of finite type over $\mathrm{Spec}\left(k\right)$. Let $X$ be a geometrically reduced $k$-variety and $Y$ a ...
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1answer
48 views

Products of Varieties

I have a few questions about products. I think I understand it but would like to hear some additional insights. i) If $X\subseteq \mathbb{A}^n$ and $Y\subseteq \mathbb{A}^m$ are affine varieties we ...
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31 views

Question about geometrical invariant

Assume $R$ is ring and $I $is ideal of $R $ The property of ideal $I$ was defined Geomerical properties which only depend on radical of $I$ For example varieties and projective varieties with ...
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0answers
27 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.
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1answer
38 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 ...
1
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2answers
95 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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3answers
184 views

Intersection multiplicity of the curves

I want to find the intersection multiplicity of the curves $f(x,y)=x^5+x^4+y^2$ and $g(x,y)=x^6-x^5+y^2$ at the point $P=(0,0)$. That`s what I have tried: $f$ and $g$ have a common tangent, the ...
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2answers
49 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
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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1answer
27 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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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 ...
3
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1answer
48 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 ...
0
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1answer
28 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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2answers
49 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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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
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) ...
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1answer
40 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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23 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 ...
3
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2answers
114 views

How to prove that sections of family of curves do not exist (exercise 4.7 in Harris' Algebraic Geometry)?

In exercise 4.7 of his book Algebraic Geometry, Prof. Harris asked to show that there is no local section of the universal hyperplane section of a smooth plane conic or the twisted cubic. The question ...
1
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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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2answers
238 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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2answers
42 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 ...
2
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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 ...
0
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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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17 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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312 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
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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 ...
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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10answers
8k views

Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
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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 ...
2
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1answer
69 views

Castelnuovo-Mumford regularity and maximal degree of generators

I am reading a few texts on Castelnuovo-Mumford regularity. If I understand correctly, almost all of them say: If $I$ is a homogeneous ideal in $k[X_0,...,X_n]$ where $k$ is algebraically closed ...
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0answers
25 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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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 ...
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46 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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39 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 ...
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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16 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 ...
2
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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
21 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 ...
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2answers
53 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 ...
0
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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 ...