The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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1answer
55 views

do the homomorphisms between two group schemes form a sheaf in the (whatever)-topology?

By this I mean: Suppose you have two group schemes $G,H$ over a scheme $S$. Then you have a presheaf on the category $\text{Sch}/S$ sending $$(T\rightarrow S)\mapsto\text{Hom}_T(G_T,H_T)$$ which is a ...
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24 views

Where to learn about the Chow scheme and the Hilbert-Chow morphism?

I would like to learn something about the Chow scheme of cycles on an algebraic variety. I am not after an abstract treatment of the moduli problem in full generality, actually I would be happy with a ...
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56 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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1answer
42 views

Invertible sheaves on affine varieties

Let $X:=\rm{Spec}(A)$ be an integral, noetherian, affine variety, and let $L$ be an invertible sheaf on $X$, I try to find an example where $L$ is not isomorphic to the structure sheaf of $X$. In the ...
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25 views

Reference request on numeric semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
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2answers
34 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
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1answer
27 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
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1answer
59 views

Blowup of a (very) simple singularity

Take the action of $\mathbb{Z}_2$ on $\mathbb{C}^2$ given by $(-1) \cdot (z,w) = (-z,-w)$ and of course, $(1)\cdot (z,w) = (z,w)$. If you look at the resulting quotient space $\mathbb{C}^2 / ...
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35 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
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39 views

An exercise in Gathmann's lecture notes about Projective spaces

Exercise 3.5.1 in Gathmann's lecture notes: Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb{P}^3$, and let $P\in \mathbb{P}^3 \setminus (L_1 \cup L_2)$ be a point. Show that there is a ...
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71 views

Only $f^\sharp_x$ makes the diagram commutative

By Algebraic Geometry I from Görtz, Wedhorn page 60 $f^\sharp_x$ is the unique ring homomorphism which makes the diagram $A\to B \to B_{p_x}$, $A\to A_{p_{f(x)}}\to B_{p_x}$ commutative. The first ...
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1answer
43 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...
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0answers
44 views

Are noetherian hypotheses necessary for the theory of the etale fundamental group?

The etale fundamental group, as explained in SGA 1 Expose 5 and various other notes I've read, always makes the assumption that the scheme $S$ (for which one intends to construct a fundamental group), ...
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40 views

Resolving the Base-points through Blow-ups

This is related to a question I asked earlier: Link So, the Hesse pencil is given by $\lambda (x^3+y^3+z^3)+\mu xyz=0$, where $[\lambda,\mu]\in\mathbb{CP^1}$ and $[x,y,z]\in\mathbb{CP^2}$. I can ...
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2answers
111 views

What are local homomorphisms, geometrically?

For want of a better name, let us say that a ring homomorphism $f : A \to B$ is local if it (preserves and) reflects invertibility, i.e. $f (a)$ is invertible in $B$ (if and) only if $a$ is invertible ...
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1answer
34 views

Matrix notation of an ellipse.

When I was reading a paper related to computer vision, I came across the following notation, where an ellipse is represented by the equation $\mathbf{x}^TM\mathbf{x} = 1$, where the ellipse parameter ...
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1answer
23 views

Ramification filtration for automorphism group of Artin-Schreier curve

I am studying the curve over the algebraic closure of $\mathbb{F}_3 = K$ defined by the equation $y^3 - y = x^4$. The automorphism group I am looking at is the one generated by elements $\sigma$ and ...
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58 views

List of exercises and examples to see the geometry behind algebraic geometry

What exercises should one solve (understanding proofs included) to gain an intuition for algebraic geometry? What are examples of (not too hard) problems that algebraic geometry handles easier than ...
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222 views

Example I.4.9.1 in Hartshorne (blowing-up)

Let $Y$ be the irreducible curve of $\mathbb{A}^2$ given by $y^2 = x^2(x+1)$. Let $t,u$ be homogeneous coordinates of $\mathbb{P}^1$. Then the total inverse image of $Y$ under the blowing-up $\phi: X ...
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Hesse's Pencil: Base Points and Resolution of Singularities through Blow-ups

A snippet of the definition given on wikipedia (full link: here) The Hesse pencil is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the ...
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2answers
139 views

Is every affine scheme the complement of the closed point $x$ of the spectrum of a local ring $A$?

Let $R$ be a commutative ring with identity element and let $\operatorname{Spec}(R)$ be the associated affine scheme. Does for each affine scheme $\operatorname{Spec}(R)$ exist a local ring $A$ ...
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2answers
78 views

Surjection from a Noetherian ring induces open map on spectra?

Let $A$ be a Noetherian ring, $f: A\rightarrow B$ a surjective ring map, then should the induced map on spectra $f^*: Spec(B)\rightarrow Spec(A)$ be an open map? In Atiyah and Macdonald, Chapter 1, ...
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95 views

Blowing up at a subvariety

Let $Y\subseteq\mathbb{A}^n$ be an affine variety with $\mathbb{I}(Y)=(f_{1},\ldots,f_{s}) \subseteq k[x_{1},\ldots,x_{n}]$. Define $\psi:\mathbb{A}^n \to \mathbb{P}^{s-1}$ by ...
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1answer
61 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
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2answers
115 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
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1answer
66 views

Blow-Up over a Field

I want to prove that a function $\pi : \mathbb{C}_{*}^{n}\mapsto \mathbb{C}^{n}$ is bijective. Where $\mathbb{C}_{*}^{n}$ is the explosion of $\mathbb{C}^{n}$ and is defined as $\mathbb{C}_{*}^{n}:= ...
3
votes
2answers
76 views

Why do only fixed points contribute to the Euler characteristic?

Let $G$ be an algebraic group with zero Euler characteristic, acting on a variety $X$ (over $\mathbb C$). I read some time ago that then the Euler characteristic of $X$ can be computed as ...
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77 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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42 views

Software tools for medium-scale systems of polynomial equations

I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...
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1answer
51 views

Base-points and invertible sheaves

Once again I am confused after thinking too much about something I thought I already understood... Let $\mathcal{L}$ be an invertible sheaf on a smooth projective curve $X$ such that $\deg ...
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1answer
87 views

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
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1answer
51 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
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28 views

Quotients of varieties by polynomial relations

Let $V$ be an affine variety in $\mathbb{C}^{n}$, i.e. $V$ is the vanishing set of an ideal $I \subset \mathbb{C}[x_{1}, \dots, x_{n}]$. Furthermore let $g \in \mathbb{C}[x_{1}, \dots, x_{n}]$. ...
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1answer
36 views

Dimensions of global sections of a divisor and its pullback

I doubt the following claim, but it seems that the proof of Theorem 10.2 (page 301, and one can download the book from libgen.org) in the book "algebraic geometry: an introduction to birational ...
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1answer
28 views

Is finiteness of rational points preserved by duality?

Sorry if this is obvious. I don't know much about Abelian varieties. Let $A/k$ be an abelian variety. Let's say $k$ has characteristic zero. Let $\widehat{A}$ be the dual abelian variety. Suppose ...
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1answer
46 views

Lifting vector bundles along thickenings

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$? The answer is yes when $X$ is affine. In general, there may be ...
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1answer
49 views

What kind of points are there in a finite type $k$-scheme?

Let $k$ be an arbitrary field and $X$ a $k$-scheme of finite type (i.e. a scheme with a finite cover of spectra of finitely generated $k$-algebras). How can I think of the points $x\in X$? What ...
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1answer
134 views

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
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0answers
108 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
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2answers
90 views

Problem I.4.7 in Hartshorne

Let $X,Y$ be varieties and suppose we have points $P \in X, Q \in Y$ such that the corresponding local rings are isomorphic, i.e. $\mathcal{O}_{Q,Y} \cong \mathcal{O}_{P,X}$. Then the problem is to ...
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Trivial Rost-Motive of a quadric

Let $q$ be the anisotropic,quadratic form of rank two corresponding to $\alpha = d(q) \in H^1(k,\mu_2)$. In his lecture notes "Topics in quadratic Forms" Vishik writes: For $n=1$ we get the ...
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Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
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34 views

Equivalence of definition of projective morphism

In Hartshonre p103, it is mentioned the two definitions of projective morphism coincide: 1)Let $f:X\to Y$ be a morphism, it is projective if it factors through a closed immersion followed with ...
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29 views

Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
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1answer
54 views

Complete intersections are connected

I have been stuck on this exercise in Vakil's notes (and moved on hoping it would come to me later), and it seems to be useful for other results (for example, when expressing curves as complete ...
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1answer
31 views

Dimension of subsets of nonsigular variety

Could you help me to prove this question? Do you have any idea? Let $X$ be a nonsingular variety and $Y \subseteq X$ is close and nonsingular . Then for any $x \in Y$ which $\dim Y_x=\dim X_x ...
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1answer
72 views

Finding the area enclosed by $px^4+qxy+ry^2+sy+t=0$

When $px^4+qxy+ry^2+sy+t=0\ (p,q,r,s,t\in\mathbb R)$ represents a simple closed curve on the $xy$ plane, can we represent the area enclosed by this curve by $p,q,r,s,t$? If yes, then how? Example 1 ...
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30 views

Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...
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2answers
93 views

geometric motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...