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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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
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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 ...
1
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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 ...
2
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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 ...
3
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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 ...
8
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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 $A$ ...
2
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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 ...
4
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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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40 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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64 views

Theorem 8.17 , Chapter II, Hartshorne

Let X be a nonsingular variety of dim n over an algebraically closed field k. Let Y be an irreducible closed subscheme defined by a sheaf of ideals $\mathscr I$. Then I want to prove that Y is a ...
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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
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 ...
2
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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 ...
4
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1answer
44 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 ...
3
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2answers
71 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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44 views

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 ...
2
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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 ...
2
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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
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 ...
2
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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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83 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 ...
1
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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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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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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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1answer
27 views

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
73 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. ...
3
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1answer
74 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 ...
0
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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 $K_X^2$ ...
3
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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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23 views

Is $U/U(w) = U \cap w U^- w^{-1}$? [closed]

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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1answer
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: ...
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26 views

Chow group of zero cycles of a product

I have been starting to learn about about chow groups. I don't know much yet, so hopefully the following is trivial: :-) For a smooth (projective, if you like) variety $X$ over a field $k$ I will ...
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1answer
51 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
3
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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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43 views

Fibers of toric morpisms

Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber ...
2
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1answer
42 views

Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
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39 views

Chow Groups of $\mathbb{P}^n$

I'm trying to see that $A_k(\mathbb{P}^n)=\mathbb{Z}$ for all $k$. I am trying to do this with induction on $n$, by applying the excision sequence $$A_k(Y) \xrightarrow{i_*} A_k(X) \xrightarrow{j^*} ...
7
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75 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
2
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1answer
46 views

Unique line through two points in projective space

I'm trying to solve exercise I.3.15 in Hartshorne's Algebraic Geometry. The question starts as follows: Projection from a point: Let $ \mathbb{P}^{n } $ be a hyperplane in $ \mathbb{P}^{n+1 } $ ...
5
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1answer
71 views

Chern classes of tautological bundle over the Grassmannian G(2,4)

I've the following problem: I know how to calculate Chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Schubert calculus. If I am right, the Chern character should ...
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1answer
64 views

Learning Fibre Bundle from “Geometry and Topology” by Bredon

Bredon defines bundle projection in the following way: Then he defines Fibre Bundle The he Remarks about the condition 3. He says the map $\theta :U \rightarrow K $ exists. The only important ...
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35 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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28 views

Exceptional locus of birational morphism is a divisor.

Let $f: V\to W$ be a proper birational morphism of smooth varieties, in a paper I'm reading the author claims that the exceptional locus of $f$ (i.e. the inverse image of the smallest closed set of ...
5
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1answer
56 views

Does pullback of schemes by monomorphism produce topological pullback?

Suppose I have scheme maps $X\to Z,Y\to Z$. It is not in general true that the fibre product $X\times_{Z}Y$ has the same underlying topological space (or even the same underlying set) as the fiber ...
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38 views

parametrization of a rational variety

Let $Y$ be a affine variety of dimension $d$ in affine space $\mathbb{A}^n$. Let the vanishing ideal of $Y$ be generated by $f_1,\dots,f_n \in A=k[y_1,\dots,y_n]$. Now suppose that $Y$ is rational. ...
5
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0answers
55 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
0
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0answers
38 views

Direct limit of $ \ \ \mathcal{D} = ((\mathcal{F} (U))_{U \in \mathcal{V}} \, \ (r \ : \ \mathcal{F} (U) \to \mathcal{F} (V))_{V \subset U}) $. [duplicate]

Let $ X $ be a topological space. Let $ \mathcal{F} $ be a sheaf on $ X $. Let $ U $ be an open subset of $ X $. Let $ \mathcal{V} $ the set of open neighborhoods of $ U $, which is the filter for ...
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18 views

What are the elements in $U/U(w)$?

Let $U$ be the maximal upper unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Then $$ U(w) = \{(a_{ij}) \in U: a_{ij} = 0, \text{ if } i<j, w^{-1}(i) < w^{-1}(j) \}. $$ My question ...
2
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1answer
61 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...