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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What are the irreducible components of $V(xy-z^3,xz-y^3,x-z^2+y)$?

I was reading the question here, and trying to fill out msteve's answer. It's not clear to me how to break up $V(xy-z^3,xz-y^3,x-z^2+y)$ into irreducible components, of which there should be two I ...
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12 views

Exercise 2.17: Algebraic curves - William Fulton

Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2$, e $\overline{X}, \overline {Y}$ the residues of $X,Y$ in $A(V)$ its coordinate ring; let $z= \dfrac{\overline{Y}}{\overline{X}}\in K(V)$. Find the pole sets ...
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6 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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22 views

What exactly is a torsion line bundle?

I am currently studying bigness of divisors, and its relationship between its numerical class and Kodaira dimension (using the aide of Lazarsfeld's Positivity in Algebraic Geometry I). I understand ...
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1answer
23 views

What is the difference between Hom and Sheaf Hom?

I'm reading Hartshorne's book, and in 3.6 he begins to go into detail about Ext and sheaf Ext, which are derived functors of Hom and sheaf Hom respectively. Let $\mathcal{F,G}$ be sheaves of ...
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16 views

Isomorphism after a base change…

Let $L\subseteq K$ be a field extension and let $X,Y$ be two $L$-schemes. Now take the base changes through $K$ and suppose that we have an isomorphism of $K$-schemes: $$X\times_{\text{Spec} ...
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1answer
37 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
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1answer
31 views

Birational map between singular variety and smooth variety

$A$ is singular and $B$ are smooth algebraic varieties. Is it possible that $A$ is birationally equivalent to $B$? (over $\mathbb{C}$)
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16 views

Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial

Let $X$ be a projective scheme over a field $k$. Let $\mathcal{O}(1)$ be an ample line bundle on $X$, then the Hilbert polynomial $P(E)$ is given by $m\mapsto\chi(E ⊗ O(m))$. The explicit polynomial ...
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1answer
14 views

Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
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23 views

Cubic surface as P$^2$ blowing up 6 points

This question is from the chapter 1 of Reid's note: Chapters on algebraic surfaces Suppose that L:(x=y=0), M:(z=t=0), and L$_5$:(y=t=0) lie on a nonsingular cubic surface X in P$^3$, define a ...
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10 views

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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29 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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16 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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22 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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27 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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18 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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35 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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16 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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30 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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25 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
43 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
33 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
30 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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21 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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24 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
44 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
56 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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45 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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26 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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15 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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1answer
47 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
36 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
41 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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44 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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34 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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22 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
40 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
56 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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0answers
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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13 views

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
102 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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24 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^{'} ...
2
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0answers
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
36 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 ...
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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 $F$ ...
2
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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 ...