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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8
votes
1answer
239 views

How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that ...
0
votes
2answers
171 views

some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...
14
votes
6answers
2k views

Reference for Algebraic Geometry

I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is ...
3
votes
0answers
129 views

Projecting Projective Curves

I've been stuck for quite a while on what is probably a trivial problem. Let $X\subset\mathbb{P}^n$ be a smooth projective curve, and let $$\mathcal{I}=\{(p,q,r):p,q\in X,p\neq ...
12
votes
1answer
612 views

Irreducible components of topological space

Let $X$ be a topological space. Let $\Sigma$ be the set of irreducible components of $X$. Let $X=\cup_{i\in I} X_i=\cup_{j\in J} Y_j$, $X_i,Y_j\in \Sigma $ for some index set $I,J$. $X_i$'s are ...
1
vote
1answer
70 views

localized ideal of a variety that is smooth at a point

Let $X \subset \mathbb{A}^n$ be an affine variety that is smooth in $a$ of dimension n-k and I its ideal in $K[x_1,...,x_n]$. Is it true that the localized ideal $I_{m_a} \subset \mathcal{O}_{X,a}$ is ...
1
vote
0answers
192 views

Algebraic Geometry and Statistics

In order to do statistical inference on a model, it must be identifiable. In other words, let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Then the model if identifiable ...
6
votes
2answers
335 views

The genus of an algebraic curve is invariant under isomorphisms

I would like to know how to prove (or even better to see a full proof) of the following "fact". Let $C_1$ and $C_2$ be two smooth curves and let $\phi : C_1 \rightarrow C_2$ be an ...
15
votes
3answers
1k views

Interesting implicit surfaces in $\mathbb{R}^3$

I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement! Some ...
3
votes
2answers
366 views

transcendental base extension

An exercise in Hartshorne claims that a scheme $X$ of finite type over a field $k$ is geometrically irreducible (respectively geometrically reduced) if and only if $X \times_k K$ is irreducible ...
2
votes
1answer
154 views

Divisors over a projective curve

I have a few doubts about some algebraic geometry problems. This is the situation: X is a (smooth) curve on the projective plane $P^2(F)$, F being an algebraic closure of $F_2$ $X = ...
2
votes
1answer
230 views

What happens geometrically when the Jacobson radical is non-zero?

I'm interested in intuition about the affine schemes of rings with a non-vanishing Jacobson radical. In the ring of real-valued continuous functions on a topological space the Jacobson radical is ...
6
votes
1answer
257 views

bijective morphism of affine schemes

The following question occurred to me while doing exercises in Hartshorne. If $A \to B$ is a homomorphism of (commutative, unital) rings and $f : \text{Spec } B \to \text{Spec } A$ is the ...
1
vote
0answers
98 views

bad behaviour of blow-up

I am studying the Blow-up contruction using the $\mathrm{Proj}$. I am looking for examples in which this construction is singular. Does anyone know situations in which the transform of the blowed-up ...
2
votes
1answer
73 views

Basic question about tautological classes (kappa classes)

Given a family of curves $\pi: C \to S$, let $\omega_\pi$ be the relative canonical sheaf over $C$. We define the kappa classes $\kappa_{i-1}$ to be $\pi_\ast (c_1(\omega_\pi)^i)$, for $i = ...
2
votes
3answers
709 views

What are the differences among an affine variety, a vector space, and a projective variety?

What are the differences among an affine variety, a vector space, and a projective variety? Are there some nice examples to explain this? Edit: For example, what is the difference between the ...
7
votes
1answer
344 views

Grothendieck connections and jets

The following question is based on some remarks in section I.2 of Deligne's book Equations Différentielles à Points Singuliers Réguliers. Let $X$ be a smooth complex variety and ...
1
vote
1answer
172 views

Pushforwards and pullbacks and projective morphisms

I am trying to understand projective morphisms (primarily from reading Ravi Vakil's Foundations of Algebraic Geometry notes Ch 17 and 18) and I have run across a very basic problem. Say I have a ring ...
4
votes
1answer
152 views

On a characterization of the tamely ramified coverings of the fraction field of a strict Henselian ring

Let $K$ be the field of fractions of a strictly Henselian discrete valuation ring $A$. Following Milne ("Etale Cohomology", Chapter I, Paragraph 5, example e)), $Spec(K)$ is the algebraic analogue of ...
1
vote
1answer
150 views

multiplication on Jacobian and corresponding map on the curve

let $X$ a smooth algebraic curve over a field , let $A$ be its jacobian and $n:A\rightarrow A$ the multiplication by $n$ map (can assume $n$ coprime with the char of the base field if this semplifies ...
14
votes
1answer
894 views

Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
3
votes
1answer
178 views

Weierstrass Equation and K3 Surfaces

Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a ...
4
votes
0answers
140 views

A question about an example on flat families from Hartshorne. In particular, is this local ring reduced?

Is the local ring $R_p$ reduced, where $p=(a,x,y,z)$ and $R=k[a,x,y,z]/I$ and $I=(a^2(x+1)-z^2,ax(x+1)-yz,xz-ay,y^2-x^2(x+1))$ ? This comes from example III.9.8.4 in Hartshorne's algebraic geometry. ...
3
votes
1answer
164 views

Tangent space of quasi-projective varieties

If $X$ is a quasi-projective variety and $X_i,\;\;i=1,\ldots,k\;$ are its irreducible components, then why $$\mathrm{dim}\;T_{X,x}=\mathrm{max}_{i=1,\ldots,k}\;(\mathrm{dim}\;T_{X_i,x})?\qquad \qquad ...
4
votes
2answers
153 views

K3 surface criteria

Suppose I have an affine equation $f(x, y) = 0$ which after homogenizing becomes $f(X, Y, Z) = 0$ in $\mathbb{P}^{3}$. Are there ways to check that $f$ represents a K3 surface?
3
votes
1answer
137 views

Irreducible subspace of $\mathbb{A}^2$

Let $X:=V(x^m-y^n)$ be a subspace of $\mathbb{A}^2$. How can I prove that if $(n,m)=1$ then $X$ is irreducible? I think that it is isomorphic to $\mathbb{P}^1$ but I can't prove that.
5
votes
3answers
332 views

Projective spaces with Zariski topology

Why $\mathbb{P}^1\times\mathbb{P}^1\not\cong\mathbb{P}^2$ where the projective spaces have the Zariski topology?
3
votes
0answers
55 views

Is any K3 surface of degree 8 in P^5 the complete intersection of quadrics?

Here the base field is the complex numbers C.
2
votes
1answer
710 views

Weierstrass Form of Elliptic Curve

One can put every cubic curve into Weierstrass form, how unique is this form?
0
votes
1answer
330 views

Codimension and Krull's principal ideal theorem

I know that in an UFD, each minimal prime will be principal. So, let $k[x_1,...,x_n]$ be a polynomial ring over a field. Further, set $S =k[x_1,..,x_n]/P$, and suppose that this is an UFD. A ...
7
votes
0answers
160 views

What's the relation between cohomology and unramified Galois covering of curves

The following statement in a paper puzzles me: "We may view $H^1(X(N), \mathbb{Z}/\ell\mathbb{Z})$ as classifying unramified Galois coverings of $X(N)$ with structure group ...
5
votes
0answers
105 views

Ramification of an integral closure of $\mathbb{C}\{z\}$

Let $\mathbb{C}\{z\}$ be the ring of convergent series in one variable over $\mathbb{C}$, $K$ the fraction field of $\mathbb{C}\{z\}$, $E$ a Galois extension of $K$ and $\mathcal{O}_{E}$ the integral ...
6
votes
3answers
643 views

Why are projective morphisms closed?

It is a well-known fact that if $X$ is a projective curve and $p \in X$ a smooth point, then any rational map $X \to Y$, $Y$ a projective variety, extends to a rational map $X \to Y$ regular at $p$. ...
2
votes
0answers
75 views

Regarding coordinate rings and subvarieties

Say that we have a polynomial ring $k[x_1,...,x_n]$ over a field k, and that J is some prime ideal of this ring, and call the quotient ring $S = k[x_1,...,x_n] / J$. Now, let K' be some prime ideal ...
1
vote
1answer
308 views

Line bundles on singular curves

assume we have a curve X of any genus with one non separating node. Call it $x$. Which is the explicit form of the boundary map $ \mathbb{G}_m\rightarrow H^1(X,\mathcal{O}_X^{*}) $ It seems to me ...
6
votes
2answers
230 views

Showing equivalence of two definitions of the Blow-Up of a Variety

Let $X=\mathop{\mathrm{Spec}}(A)$ be an affine variety over some algebraically closed field $\Bbbk$ and $I\subseteq A$ an ideal of $A$. There are two ways to define the blow-up $\tilde X$ of $X$ along ...
5
votes
2answers
753 views

I want a proof without using Nakayama's lemma

I am trying to understand Nakayama's lemma. It looks like some "fixed point theorem". Using Nakayama's lemma , I can easily solve the following question. I want another proof. Thanks. Let $A$ be a ...
7
votes
2answers
191 views

effective Cartier divisor is trivial

Given a schema $X/k$ with $H^0(X,\mathcal{O}_X^\times) = k^\times$ and an effective Cartier divisor $D \geq 0$ such that $\mathcal{O}(D) = O_X$, why is necessarily $D = 0$? I tried to apply the long ...
4
votes
0answers
119 views

Hecke operators as endomorphism of Jacobians of modular curves

Let $p$ be a primes that does not divide $N$, then $T_p$ defined an endomorphism $J_0(N)\to J_0(N)$. what is $T_p^\vee$? In other words, we naturally have $J_0(N)^\vee \xrightarrow{T_p^\vee} ...
2
votes
1answer
168 views

Matrices of rank and singular points

I am stuck on the following homework problem: Show that if S=S(m,n,r) represents the space of m by n matrices with rank less than or equal to r (naturally isomorphic to an affine subvariety of ...
5
votes
3answers
254 views

How to imagine $X^2+Y^2-1=0$ in $C^2$, $X$, $Y$ both complex

Like the title, with a graph if convenient, I was reading Mumford's book, a picture make me confused: Could someone explain it to me, thanks.
7
votes
2answers
268 views

Finite etale maps to the line minus the origin

I am trying to determine the etale fundamental group of $V = A^1 - \{0\}$ over an algebraically closed field $k$. I am trying to stay in the comfortable zone of non-singular varieties. To do this, I ...
0
votes
1answer
244 views

Does this correctly show that the union of infinite affine varieties is not an affine variety?

The union of two affine varieties can be expressed as $$ \mathbb{V}(\{F_i\}_{i\in I}) \cup \mathbb{V}(\{F_j\}_{i\in J}) = \mathbb{V}(\{F_iF_j\}_{(i,j) \in I\times J}). $$ We want to generalize this ...
3
votes
3answers
249 views

Conjugation is not expressible in terms of polynomials

In order to convince myself that the set $U(n)$ of unitary matrices (matrices with columns that are orthonormal under the complex inner product) is not an affine variety in $\mathbb{C}^{n^2}$, I need ...
3
votes
2answers
121 views

Showing a (relatively simple) set of polynomial zeros in projective space is irreducible

I'm teaching myself a little algebraic geometry and I was hoping you could help me with an exercise. I have my head around affine spaces alright but I am having a little more trouble with projective ...
0
votes
0answers
204 views

Could anyone please give me some hint on relation between number theory and algebraic geometry,with solid examples please!

I studied math all by my self for a few years out of my own love for it, since without guide, I was often puzzled. I'd like ask for aid here, mainly about why we view numbers over spec Z etc. Does it ...
4
votes
2answers
210 views

Components of algebraic varieties

Sorry, but I have to ask a dumb question: Algebraically, a hyperbola has only one irreducible component (given by an irreducible polynomial). Why, then, does the real image of a hyperbola show two ...
6
votes
1answer
170 views

Is there a universal property of $\text{Spec}(-)$?

I've heard it been said that the construction of Spec$R$ is a canonical way of taking the ring $A$ and producing a locally ringed space with $A$ as the ring of global sections. This is certainly ...
0
votes
1answer
161 views

find skew lines on a cubic surface for a parametrization

I consider the hypersurface $Y = V(y(x^2+z^2)-x) \subset \mathbb{A}^3_k$ ($k = \mathbb{C}$) I've read that if you have two skew lines on a non-singular cubic surface $Y$, given by a polynomial of ...
2
votes
2answers
235 views

morphism of the local rings correspond to what kind of maps between varieties

To a regular(or polynomial) map $f: X \to Y$ between affine varieties we associate its pullback $f^\ast: K[Y] \to K[X]$ and it holds that f is an isomorphism iff $f^\ast$ is an isomorphism. Now if ...