All questions with tag [math: algebraic-curves]


projective cubic

I have some troubles to confirm that the photo of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set $$V(XT-YZ,\, Y^2-XZ,\,Z^2-YT)$$ Clearly I have trouble to confirm that the algebraic projective set is had in the photo of $f$. Specifically "solv...
2022-07-25 17:16:00

The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ connected) is isomorphic to the finite etale morphism $X\to X$ given by $z\mapsto z^n$ for some $n\geq 1$. The universal covering space $\widetilde{X}$ of...
2022-07-25 12:45:59

Family of curves (in algebraic geometry)

How can I watch, in algebraic geometry, a family members of contours over a base contour? As an example, can the family members $y^2 = x(x-1)(x-t)$ be considered as a family members over $\mathbb{P}^1$? Just how can I recognize this carefully? Can a person aim me to a reference?
2022-07-25 12:34:39

Gaps in the Genera of Space Curves

We found out the adhering to partnership in between the level and also category of plane curves in my algebraic geometry course: \begin{array} a \text{degree} &d &1 &2 &3 &4 &5 &6 &7 & \dots\\ \text{genus} &g &0 &0 &1 &3 &6 &...
2022-07-25 07:46:46

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an affine algebraic set and calculate $I(C)$. Show that the affine algebra $Γ(C)=k[X,Y,Z]/I(C)$ is isomorphic to the ring of polynomials $k[T]$. I found a...
2022-07-25 07:41:23

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the research of algebraic contour is techniquely equivalent to the innovative department of analytic geometry, otherwise, what is the distinction? And also what is various other branch of innovative analytic geometry called? on top of that, what is the differece in between algebraic geometry and also algebraic curves?
2022-07-24 06:35:22

Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$. Let $K$ be a number field and let $X$ be a curve over $K$. Define $f_K(X)$ to be the value of $f$ after base change to $\overline{\mathbf{Q}}$. Now, a priori, this function $f_K$ is not well-de...
2022-07-24 06:22:05

Ramification on hyperelliptic curves

I am using Rick Miranda's book "Algebraic curves and Riemann Surfaces" to try and check some things about hyperelliptic curves. I have completed almost all of one of the exercises, but there is one part of my proof I am not sure of. The question (question R) 2), page 245), is as follows: Let $X$ be an algebraic curve of genus $g\geq 2...
2022-07-24 03:23:09

Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection with the continuous cohomology pointed set $$H^1(\mathrm{Gal}(\overline{K}/K),X).$$ I was just wondering about an analogous question for ...
2022-07-22 15:46:01

Artin-Schreier extensions over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do this I have viewed the curves as the function field $k(x,y)$, originally restricting to those defined by $$ y^2 - y = f(x), \ f(x)\in k[x]. $$ Aft...
2022-07-22 15:34:46

Jacobian of a curve

Let $C$ be a contour and also $J$ be its Jacobian. What is the relationship in between $H^1(C,\mathcal{O}_C)$ and also $H^1(J,\mathcal{O}_J)$? Can a person aim me to a very easy reference for this topic?
2022-07-22 15:32:42

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, w_{2}, w_{3}$ also rationals points in $S_{L|K}$, distinct from each other. I have to prove that there is a unique automorphism $\sigma$ of the exte...
2022-07-22 15:27:30

Rational curve cover/Transcendental Galois field extension

Suppose the rational curve $C$ is a finite cover for the rational curve $D$ and the field of rational functions of $C$ is the purely transcendental extension $k(x)$ and that of $D$ is the subfield $k(t)$ where $t$ is a rational function in $x$. Assume the field extension is Galois so that there is a finite group $G$ acting on $k(x)$ with the sub...
2022-07-22 15:19:44

Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the strip $$\{x+iy: -1\leq x \leq 1\}?$$ Is the following a correct argument? The matrix $$\left( \begin{matrix} 1 & \pm 2 \\ 0 & 1 \end...
2022-07-20 14:48:41

Are elliptic curves also Galois covers of degree 3

Let E be an elliptic contour with formula $y^2=x^3+Ax+B$. The estimate onto the $x$ - coordinate is a Galois morphism of level $2$. Yet what concerning the estimate onto the $y$ - coordinate? Is it Galois of level 3? Where does one research this map?
2022-07-17 14:36:24

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a limited type, apart yet non - correct morphism of schemes. Can there be a projective contour $g:C\to S$ and also a shut immersion $C\to X$ over $S$? Simply to be clear: A projective contour is a smooth projective morphism $X\to S$ such that the geometric fibers are geometrically attached and also of measurement 1. In strai...
2022-07-17 12:17:11

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ and has gonality $\leq 3$ (consider the projection). I'm trying to prove that it has gonality $3$. More generally, what is a computationally feasible ...
2022-07-16 14:37:52

Parameterizing a rational curve

I'm having trouble finding a parameterization for the following curve: $x^4 - 2x^2yz + y^2z^2 - y^3z = 0$ taken to be a curve in $\mathbb{C}\mathbb{P}^2$. I followed the example on Wikipedia where they parameterized a circle's equaton, ie., I demohogenized the polynomial at $z$, so I reduce to the curve $x^4 - 2x^2y + y^2 - y^3 = 0$. Then I obse...
2022-07-16 14:26:50

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if $K=\mathbf{Q}$. (My method of proof is a bit awkward.) I expect the answer to be no (unfortunately) in general. What is an equation for the normalization of...
2022-07-16 14:18:22

self-intersections in a product of two curves

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how does one prove this? I understand that the self-intersection is defined by intersecting $\Delta$ with a general divisor linearly equivalent to $\Del...
2022-07-15 02:45:11