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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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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20 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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10 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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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
20 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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26 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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38 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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34 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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28 views

Some questions on Hartshorne I.7: intersections in projective space

I am reading I.7 of Hartshorne, and here are some questions I don't understand. 1) Prop. 7.4. Let $M$ be a finitely generated graded module over a noetherian graded ring $S$. Then there exists a ...
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34 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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scheme-theoretic image behaves nicely with composition, base change?

Scheme-theoretic image is still somewhat of a mystery to me, and I wasn't able to work out proofs of either of the following two statements that seem plausible to me: If $X\to Y\to Z$ is a map of ...
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66 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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51 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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99 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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37 views

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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43 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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56 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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113 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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63 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}:= ...
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1answer
50 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
33 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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67 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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36 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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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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1answer
22 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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1answer
77 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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37 views

Factorization of ideals in a coordinate ring (Dedekind domain)

Consider $f \in \mathbb{C}[X,Y]$ an irreducible curve non singular. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$ and choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common ...
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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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49 views

Theorem 8.17 , Chapter II, Hartshorne

I don't understand two things in the converse part of this theorem- First of all I don't understand why $ dx_1, dx_2,...,dx_r $ generate a free subsheaf of rank $r$. Why should that subsheaf be ...
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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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50 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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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
40 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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68 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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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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2answers
89 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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27 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
27 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
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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73 views

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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1answer
30 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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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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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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Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
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1answer
71 views

Invertibility of a polynomial map equivalent to a condition on the ideal generated by the coordinates?

Let $k$ be an algebraically closed field. Let $$F=(F_1,\ldots,F_n) \colon k^n \to k^n$$ be a polynomial map. I'm trying to understand the relation between the conditions: $F$ is invertible. ...
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73 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
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15 views

canonical divisor and self-intersection number

Let $X$ be a Chatelet surface over $\mathbb{Q}$ whose affine model is given by $y^2 + z^2 = P(x)$, where $\deg(P) = 4$, and let $K_X$ be the canonical divisor. How can I compute $K_X$ and $K_X^2$ ...
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59 views

Kähler differentials of the cuspidal cubic

I want to compute $\Omega^1_{A,\mathbb{C}}$ for $A = \mathbb{C}[X,Y]/(Y^2 - X^3)$, or more precisely, I want to show that the module of Kähler differentials is free of rank 2 at the origin, and free ...
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Is $U/U(w) = U \cap w U^- w^{-1}$? [on hold]

Let $U$ be the maximal upper unipotent subgroup of $GL_n$ and $U^{-}$ maximal lower unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Is $U/U(w) = U \cap w U^- w^{-1}$? Thank you very much.
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56 views

Cardinality of variety

I'm trying to show that the cardinality of any variety of positive dimension is $ |k |$ where $k $ is the field being considered. This is part of exercise I.4.8 in Hartshorne's Algebraic Geometry: ...