Tagged Questions

5
votes
0answers
48 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
2
votes
0answers
40 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
2
votes
4answers
157 views

For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
4
votes
3answers
54 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
6
votes
1answer
46 views

Existence of a variety with prescribed properties

In these notes that give a proof of the Weil conjectures for curves, the author writes on page 17 that given a smooth projective curve $X$ over a finite field $k = \mathbb{F}_q$ for a fixed prime $q$, ...
2
votes
0answers
64 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
5
votes
1answer
32 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
9
votes
2answers
122 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
2
votes
1answer
39 views

Extension of prime ideal in $k[V]$ to $\mathcal{O}_P(V)$ is prime?

Let $k$ be an algebraically closed field, $I\subset k[X_1,\cdots, X_n]$ be a prime ideal, $V=V(I) \subset \mathbb{A}^n$ a variety and $P=(a_1,\cdots, a_n)\in V.$ Recall that $\mathcal{O}_P(V)$ is the ...
3
votes
0answers
33 views

Computing a rational function at a point in terms of a uniformising parameter

I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
11
votes
2answers
294 views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
1
vote
0answers
40 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
1
vote
0answers
54 views

A funny condition for ampleness on a curve

Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$. Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
8
votes
1answer
82 views

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
1
vote
0answers
28 views

Homogeneity of translated polynomial

I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9): The multiplicity of the complex ...
4
votes
1answer
43 views

plane cubic with a singularity must have non-constant morphism from $\mathbb{P}^1$?

If $C$ is a plane projective curve which is defined by an irreducible homogeneous cubic polynomial and has a singularity, why must there be a nonconstant morphism $\mathbb{P}^1\rightarrow C$? (I'm ...
4
votes
1answer
52 views

n-canonical embedding

Let $C$ be a stable curve of genus $g>1$ and let $ \omega $ be its dualizing sheaf. Let $n$ be a integer larger than 2. Does anyone knows how to show that $\omega^{\otimes n}$ separates points and ...
6
votes
0answers
64 views

A Regular Map Has Finitely Many Ramification Points

Let $C,D$ be nonsingular projective curves, $f \colon C \to D$ nonconstant, $K = k(C), L = k(D)$, $d = \text{deg }f = [K:L]$, and of course $k$ algebraically closed. Furthermore let's suppose that ...
2
votes
1answer
59 views

Elliptic curve as an intersection of quadrics

Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
6
votes
1answer
82 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
7
votes
2answers
108 views

Are endomorphisms of degree one always automorphisms

Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one. Do I understand correctly that $\sigma$ is an automorphism? I believe this ...
5
votes
1answer
50 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
6
votes
2answers
114 views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
5
votes
1answer
42 views

Why are finite unions of algebraic sets algebraic?

Suppose $F,G$ are polynomials in $k[x_1,\cdots, x_n]$ (k is a field). Let $$V(F) = \{ (a_1,\cdots,a_n)\in k^n : F(a_1,\cdots,a_n)=0 \}.$$ Then $V(F)\cup V(G) = V(FG),$ essentially because integral ...
2
votes
1answer
85 views

Reference request: Construction of $M_{1,0}$

Does anyone know a reference for the construction of the (Artin) stack $M_{1,0}$ and a result about the corresponding coarse moduli space? In Deligne-Mumford they construct $M_{g,0}$ when $g\geq 2$ ...
3
votes
1answer
45 views

Birational Maps of Nonsingular Projective Curves

I'm trying to solve exercise I.6.7 in Hartshorne, stated (in part) here: Let $P_1,\ldots,P_r, Q_1,\ldots,Q_s$ be distinct points of $\mathbb{A}^1$. If $\mathbb{A}^1-\{P_1,\ldots,P_r\}$ is ...
2
votes
1answer
55 views

Computing the intersection divisors of a smooth projective cubic curve.

Let $X$ be the smooth projective plane cubic curve defined by $y^2z=x^3-xz^2.$ Compute the intersection divisors of the lines defined by $x=0,y=0,$ and $z=0$ with $X$. Here is an idea: Any point ...
2
votes
1answer
91 views

Compute the principal divisors of a hyperlelliptic surface.

Let $X$ be the hyperelliptic surface defined by $y^2 = x^5-x.$ Note that $x$ and $y$ are meromorphic functions on $X.$ Compute the principal divisors div($x$) and div($y$). We have the ...
4
votes
1answer
123 views

Computing the divisors of a meromorphic function defined by a hyperelliptic curve.

Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$. I ...
3
votes
1answer
41 views

Does this diagram of Chern classes and push forwards commute

Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
3
votes
1answer
40 views

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
7
votes
2answers
111 views

Singularities of Curves in Positive Characteristic

Given a collection of polynomials $\mathscr{F}\subset\mathbb{Z}[x_1,\ldots,x_n]$, we can associate to each prime ideal of $\mathbb{Z}$ an affine variety as follows: $$ (p)\longmapsto ...
0
votes
1answer
75 views

Degree of sum algebraic functions

This question I have asked on mathoverflow already: http://mathoverflow.net/questions/123921/degree-of-sum-algebraic-functions Let $C$ - curve, $f_1, f_2 \in K(C)$. How to prove that deg$(f_1 + f_2) ...
5
votes
1answer
77 views

How much do I need to learn before I can read about Toric varieties?

I have a copy of the book "Introduction to Toric varieties" by William Fulton, and over the next few months I'd like to make some progress on it. As a first goal, I'd like to be able to read just ...
6
votes
1answer
86 views

Integral Closures and Affine Curves

Let $C$ be an irreducible affine curve with singular points, and let $A$ be its ring of regular functions. Since $C$ has singular points, $A$ is not integrally closed in its field of fractions, $K$. ...
3
votes
0answers
36 views

Spectrum theorem for p-adic matrix analysis

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\ 2: ...
0
votes
0answers
39 views

$C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
5
votes
1answer
96 views

hyperelliptic curve on abelian variety

There is serious flaw in the following argument, but I have yet to see what it is: Let $C$ be a genus $g$ hyperelliptic curve in an abelian variety $A$ (over an algebraically closed field with ...
4
votes
2answers
136 views

Every curve in $\mathbb{A}^3$ is the zero-locus of $3$ polynomials

I have the following problem: Let $X \subset \mathbb{A}^3$ which doesn't contain a vertical line and let $g \in K[x,y]$ such that $g$ hasn't any double factor and the clousure of the projection of ...
6
votes
2answers
89 views

Principal divisors

How can i calculate the principal divisor $(f)$ donde f es: $\displaystyle\frac{(x^{3}-1)}{(x^{4}-1)}$ with $f\in\mathbb{F}_2(x)$ I am recently reading about the subject, so i am looking for a simple ...
3
votes
1answer
66 views

Factorization over a relatively minimal surface

Let $A$ be a DVR with an algebraically closed residue field $k$. Consider a morphism $f: X \to Y$ of arithmetic surfaces (regular integer projective and flat schemes over $A$ of dimension 2), such ...
4
votes
0answers
50 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
5
votes
1answer
156 views

Basis for the Riemann-Roch space $L(kP)$ on a curve

Let $C$ be a smooth algebraic plane complex curve of degree $d$ defined as the zero locus of a homogeneous polynomial $F$. Let $P\in C$. For a positive integer $k$ consider the divisor $$D=kP$$ ...
0
votes
0answers
36 views

Is the number of automorphisms of a hyperelliptic curve bounded

Certainly, if we fix the genus $g$ of a curve $X$, we have $\# $Aut$(X) \leq 84(g-1)$. Let $X$ be a hyperelliptic curve. Is there a bound on $\#$Aut$(X)$? (Note that I do not want to fix the genus!) ...
2
votes
0answers
32 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
6
votes
2answers
93 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...
1
vote
0answers
38 views

Why is $\kappa_1$ ample?

Let $\pi:C_g=\overline M_{g,1}\to \overline M_g$ be the "universal curve". I recall that on $C_g$ there is the line bundle $L$ defined by $L_{(C;P)}=\Omega^1_{C,P}$, giving rise to $\psi:=c_1(L)\in ...
1
vote
0answers
34 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
2
votes
0answers
19 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
3
votes
1answer
103 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...

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