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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Ring of integers in a Artin-Schreier extension

It is well-know( see Goldschmidt book: Algebraic Functions and Projective Curves) that a for $q$ a power of $2$ a quadratic separable extension of $\mathbb F_q(T)$ can be written as: $$K:=\mathbb ...
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12 views

Vector bundle of rank 2 that is not an extension of two line bundles

Let $X$ be a (smooth) projective surface over a field. Is true that for any such $X$ there is a vector bundle $F$ of rank 2, that is not an extension of two line bundles? That is $F$ can not be used ...
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11 views

Closed slice of affine space has closed projection

Let K be an algebraically closed field and suppose a set $\{ \overline{x} \} \times A$ is closed in the Zariski topology of the affine space $K^m \times K^n$ ($\overline{x} \in K^m, A\subset K^n$). ...
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15 views

Proof of Chevalley–Warning theorem

How to prove Chevalley–Warning theorem (http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem) by using Fulton's trace formula (#$|X(\mathbb{F}_p)| \equiv \sum (-1)^i Tr(Frob_p|H^i(X, ...
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24 views

on the Projective Nullstellensatz

In his book Algebraic Curves, Prof. Fulton says the following with regard to the projective Nullstellensatz: I am not sure what he means by "to make an exception" with respect to the irrelevant ...
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14 views

What is the best book to learn coordinate geometry

The level should be above high school, and it must be free online if at all possible. Also, I have an additional question to ask: How many months, roughly, would it take to finish a mathematics book ...
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1answer
29 views

An example of a family of projective irreducible curves

I'd like to construct explicitely (namely with a parametric equation) the following example: A family of projective curves parameterized by $\mathbb P^1(\mathbb C)$ with 3 properties: All curves ...
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14 views

Conditions for principal divisors

If $C$ is a complex riemann surface and $D$ a general divisor on $C$. Is there a way to say that this divisor is principal, i.e. $D$ is the divisor of a meromorphic function on $C$? I ask this ...
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24 views

How to write a divisor

Let $C$ a complex Riemann surface (compact) and $\alpha:C^{'} \rightarrow C$ an unramified cover of $C$. Define the application $\alpha_{*} :Div(C^{'}) \rightarrow Div(C)$ as follow $$\forall E\in ...
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23 views

Involution on divisor

Suppose that $\pi:C^{'} \rightarrow C$ is an unramified double cover of a complex riemann surface. Let $\tau$ the involution sheet exchange. If $E$ is a divisor on $C^{'}$ What is the definition of ...
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1answer
38 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: A point on the ellipse The slope of the tangent line to the ellipse at that point Another point ...
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1answer
31 views

On a curve $\mathcal{O}(p)\cong \mathcal{O}(q)$ implies the curve is $\mathbb{P}^1_k$ in Ravi Vakil's notes

In Ravi Vaki's notes he proves then if a curve (projective, geometrically regular/integral) over field $k$ has a degree one line bundle with two or more global sections, then that curve is isomorphic ...
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1answer
29 views

Two questions about discrete valuation rings of varieties

Let $X$ be a proper, normal variety over $\mathbb{C}$, and $k(X)$ be its field of rational functions. I think the following two statements are true, but I was unable to give a proof or find the ...
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20 views

Two ideals that agree in the completion [on hold]

Suppose that $R$ is a Noetherian local ring with maximal ideal $\mathfrak m$, and that $I$ and $J$ are two ideals in $R$ with $\hat{I} = \hat{J}$ in the completion of $R$ at $\mathfrak m$. What can ...
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23 views

Valuative criteria at closed points

Let $X \to S$ be a morphism. In the valuative criteria for properness, is it enough to test morphisms $\text SpecK \to X$ from spectra of fields to $X$ such that the image is a closed point of $X$? ...
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2answers
41 views

hyperbolic geometry and affine transformations

A very famous geometer recently commented to me that "hyperbolic geometries are the only geometries invariant under affine transformations". It is unclear to me what this comment even means. Can ...
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1answer
45 views

Are vector bundles on toric varieties also toric varieties?

Let $X$ be a toric variety, and $\pi:E\to X$ a vector bundle, say of rank $2$. You can think of $X=\mathbb P^1$. Question. When is the total space of $E$, or of $P(E)$, a toric variety? What do I ...
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44 views

how to prove that two sets have the same homotopy type

I want to prove that two bounded sets $A$ and $B$ of $R^2$ have the same Euler characteristic (number of connected components minus number of connected components of the complementary) but I think ...
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25 views

Quadratic Transformation to make a point simple

This is exercise 7.21 from Fultons "Algebraic Curves". Let $X$ be a nonsingular projectiv curve, $P \in X$. Show that there is a projective plane curve $C$ with only orinary multiple points and a ...
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14 views

Torsion freeness is not affine local

I am working on an "unimportant" exercise (c.f. vakil, exercise 13.5.J) which goes as follows Exercise 13.5.J: Find an example on a two-point space showing that $M:=A$ might not be a torsion-free ...
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30 views

Not every complex manifold is of the form $X_{an}$ where $X$ is some algebraic variety

I want to show that not every complex manifold is of the form $X_{an}$ where $X$ is some algebraic variety, providing a counterexample. The candidate for this counterexample seems to be the open ...
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1answer
33 views

Mazur's theorem-abelian torsion group of rational points of an elliptic curve

I am looking at Mazur's theorem... $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for } n=1,2, \dots ,10,12$$ means that the torsion group $E(\mathbb{Q})_{\text{torsion}}$ ...
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1answer
40 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
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1answer
28 views

A prime ideal with the algebraic set reducible

In "Algebraic Curves" by Fulton, section 1.7, page 11, there is the following corollary of the nullstellensatz: Corollary 2: If $I$ is a prime ideal, then $V(I)$ is irreducible. There is a ...
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41 views

27 Lines on a Cubic

In Ravi Vakil's notes, there is a proof (in section 27, of course) of the famous result that every nonsingular cubic hypersurface in $\mathbb{P}^3_k$ over an algebraically closed field $k$ has exactly ...
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32 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. $\textbf{Shafarevich definition}$ (pg 128) - A variety is $\textit{non-singular in codimension one}$ if the singular locus has ...
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21 views

When using the square method , how do you know what level equation you are working with? [on hold]

When using the square method , how do you know what level equation you are working with?
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1answer
37 views

Is $\mathbb{C}[T \times_T T] = (\mathbb{C}[T] \otimes \mathbb{C}[T])^T$?

Let $T$ be an algebraic group. There is a left $T$ action on $T$ given by left multiplication and a right action on $T$ given by right multiplication. Let $T$ acts in the middle of $T \times T$ by the ...
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3answers
55 views

Counting the number of $\mathbb{F}_q$ points on a homogeneous polynomial

This is an area of number theory that I am not too familiar with and I would appreciate any assistance! Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic not 2 or 3. I have the ...
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22 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
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1answer
95 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
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1answer
32 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
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1answer
21 views

Is $\mathbb{C}[T \times_T T] \cong \mathbb{C}[T]$?

Let $T$ be an algebraic group. Do we have the following result: The algebra $\mathbb{C}[T \times_T T]$ is equal to the image of the co-multiplication $\mathbb{C}[T] \to \mathbb{C}[T] \otimes ...
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23 views

functions on Riemann sphere

Let $C$ a complex riemann surface (compact). Take $f$ a meromorphic funtion on $C$. This function defines a function $\pi:C \rightarrow \mathbb{P}^1_{\mathbb{C}}$. How can i define a function $f^{'} ...
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34 views

About the smoothness of a non-reduced variety.

Consider a $k$-scheme $X$ with the following properties: $k$ is algebraically closed, $X$ is irreducible, non-reduced, separated and of finite type; moreover $X$ has dimension $1$ ($X$ is a non ...
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1answer
33 views

How to show that a polynomial maps an algebraic set to an algebraic set?

Sorry if this is an ignorant question. I am studying algebraic geometry. This isn't an exercise problem. It is an assumption I can use to prove something else. I think it must be obvious, but I don't ...
2
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1answer
25 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
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1answer
33 views

Do we have $\mathbb{C}[T] = \mathbb{C}[\Lambda] = \oplus_{\lambda \in \Lambda} \mathbb{C}[\lambda]$?

Let $\mathbb{C}[T]$ be the coordinate ring of a torus $T$. Suppose that $T$ acts on some variety $X$. Then $T$ acts on $\mathbb{C}[X]$: $t(f) = \lambda(t)f$ ($f$ is a homogenous function on ...
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51 views

Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
2
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1answer
47 views

Universal property of sheafification

Given a presheaf $\mathcal{F}$ there is a sheaf $\mathcal{F}^+$ and a morphism $\theta: \mathcal{F}\to\mathcal{F}^+$ with the property that for any sheaf $\mathcal{G}$ and any morphism $\varphi: ...
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46 views
+50

Cohomology with coefficient $\mathbb{Q}(n)$

What is the definition of $$\mathrm{H}^i(X,\mathbb{Q}(n))$$ for a variety $X$? and What is its relation with $\mathrm{Ext}^*(\mathbb{Q}(0),\mathbb{Q}(n))$? Another question: Is this a notion which ...
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33 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
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2answers
35 views

Checking normality for quasi compact schemes

Let $X$ be a quasi compact scheme. We know that any point on $X$ is a generization of a close point. Could someone possibly explain me why it then follows that to check if $X$ is normal, it suffices ...
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1answer
40 views

Simple question about the definition of divisor

Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define the divisor $\pi^{*}(E)$?
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1answer
38 views

Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
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63 views

Finite groups and manifolds

my question is: can we connect finite groups with algebraic manifolds as: Take for example Togliatti surface $X$ with set of 31 singular points $P$. Then consider action $Aut(X)$ on $P$, then group ...
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2answers
49 views

Cremona group of $\mathbb{P}^n$

I know that the complex conjugation $\tau: \mathbb{P}^n \mapsto \mathbb{P}^n$ that sends any point $x$ to the point with complex conjugate coordinates $\tau(x)$ is a homeomorphism. In order to show ...
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1answer
23 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
3
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0answers
20 views

Coordinates on a Richardson variety

I'm looking for convenient coordinates to use to describe the intersection of a Schubert cell $X^\circ_\lambda$ and an opposite Schubert cell $\Omega^\circ_\mu$ in a Grassmannian $G(k,n)$. Describing ...