An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...

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83 views

$H^1$ of a constant sheaf

Let $X$ be an irreducible smooth curve, and $\underline{k(X)}$ the constant sheaf on $X$ with the function field $k(X)$ as fibers. Reading from Serre's Algebraic groups and class fields I met the ...
2
votes
1answer
96 views

Linear Equivalence of Divisors on Projective Plane Cubic

I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors: Let $X$ be the projective plane cubic ...
3
votes
2answers
115 views

Finding a Uniformizer of a Discrete Valuation Ring

Suppose I have a discrete valuation ring. Then what are some techniques for explicitly finding a uniformizer? I'm especially interested in situations where the ring is given similarly to the following ...
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42 views

Morphisms of affine line into a curve of higher genus

Let $X$ be a smooth curve of genus $\geq 1$ over a field $k$. How does one show that any $k$-morphism $f: \mathbb A^1_k \to X$ has to be constant? In general, is it true that any morphism $Y \times ...
3
votes
1answer
66 views

Coordinate ring of an open set

I'm trying to solve Exercise 3.1 (b) in Hartshorne's Algebraic Geometry. I see a solution of it and it says that Any proper open set of $\mathbb A^1$ is $\mathbb A^1-S $, where $S$ is a finite ...
3
votes
1answer
48 views

If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
5
votes
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66 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
5
votes
1answer
73 views

Normal curve after base change (p > 0)

Suppose that $C$ is a normal projective curve over some base field $k$, possibly of positive characteristic. I am wondering to what extent one can modify the base field $k$ while not braking the ...
1
vote
1answer
38 views

Generators of the first singular homology group of a Riemann surface

Let $X$ be a Riemann surface embedded in $\mathbb C^2$ with coordinates $(z,w)$ and let $\pi_z \colon X \to \mathbb C$ be the projection on first coordinate with property that that for cofinite number ...
5
votes
1answer
150 views

Tangent sheaf of the Picard scheme

Let $X\overset{f}\to S$ be a flat morphism of schemes, and $S=\operatorname{Spec}(k)$ for an algebraically closed field $k=\bar{k}$. I'm just interested, for the moment, with schemes $X$ corresponding ...
3
votes
1answer
81 views

Divisors on a complex torus

I'm asked to prove the following fact: On a complex torus $X$ every canonical divisor is principal and vice-versa. At this moment I know only the basic properties of divisors and that, if $K$ is a ...
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0answers
58 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
5
votes
1answer
71 views

Given a non-singular curve $C$, show that two divisors are algebraically equivalent iff they have the same degree

I wish to show that given a non-singular curve $C$, two divisors are algebraically equivalent if and only if they have the same degree. I'm rather stuck on how to approach such a problem. I'm ...
2
votes
1answer
109 views

Plane algebraic curves in $\mathbb C^2$ are connected in the analytic topology.

Is there a "simple" proof, not involving much tools of Algebraic Geometry, to the fact that every irreducible affine curve $C=\{(z,w)\in\mathbb C^2\,:\, F(z,w)=0\}$ (where $F\in\mathbb C[X,Y]$ is ...
2
votes
1answer
54 views

Finite etale covers of pro-curves

Let $X$ be an inverse limit of integral, normal affine curves $X_i = Spec(A_i)$ where all the transition maps are finite etale covering maps (so in particular $X$ exists in the category of schemes), ...
5
votes
0answers
83 views

Curves of fixed genus and degree lying on a cubic surface

I would like to prove the following statement: Let $C\subseteq \mathbb{P}^{3}$ be an irreducile nonsingular curve of arithmetic genus $g_{a}(C)=24$ and degree $d(C)=14$. Then there exists an ...
3
votes
0answers
64 views

Divisors as a scheme

Let $S$ be a scheme and $X$ be a scheme over $S$ corresponding to a smooth projective curve. Define $X_d := X^d / \mathfrak{S}_d$, where $X^d$ denotes the usual $d$-fold cartesian product of $X$ and ...
4
votes
0answers
64 views

Algebraical Fujita cancellation

If there exists some birational equivalence between ruled surfaces $C\times\mathbb{P}^1$ and $C'\times\mathbb{P}^1$ over a sufficiently nice field, then one can cancel the projective line and conclude ...
4
votes
1answer
180 views

Why does the degree of a line bundle equal the degree of the induced map times the degree of the image plus the degree of the base locus?

Let $L$ be a line bundle on a smooth curve $C$. If $L$ is rank $r+1$, define the induced map (as Arbarello, Cornalba, Griffiths, Harris): $$\begin{aligned}\phi :& C \rightarrow \mathbb ...
2
votes
1answer
48 views

Irreducible algebraic sets with intersecting parts

Let $V = V(F)$ be an irreducible hypersurface in $A^n(k)$. To show: If $W$ is an irreducible algebraic set in $A^n(k)$ with $V \subset W$, then $V = W$. The ideas I got so far: Since $V, W$ are ...
4
votes
1answer
183 views

Exercise 3.18 of Fulton's Algebraic curves.

I'm trying to provide a proof of the following fact: If $p$ is a simple point on the curve $F$ then $I(p,F\cap G)=ord_p^F(G)$. Where $I(p,F\cap G)$ denotes the intersection number of the curves at ...
3
votes
0answers
71 views

Parametric 12-deg and 14-deg equations with group $PGL(2,11)$ and $PGL(2,13)$?

We have, $$x^{12} - a x^{11} - 33x^8 + 22a x^7 - 11a^2 x^6 + 363x^4 - 121a x^3 + 121a^2x^2 - 23a^3x - 11^3 + a^4=0$$ $$x^{12} - a x^{11} - 11a x^9 - 44a x^7 - 88a x^5 - 88a x^3 - 32a x - a^2=0$$ ...
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45 views

Specifying the real multiplication on a curve

Suppose we have a curve $C$ of genus two over an algebraically closed field of characteristic zero with a given Weierstrass point $p_0$, furthermore we assume this curve to have real multiplication by ...
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0answers
42 views

Does this proof that the resultant provides an upper bound for intersection multiplicity look correct?

Let $f,g \in \mathbb{C}[x,y]$ such that $f(0,0)=g(0,0)=0$ and the varieties $V(f)$ and $V(g)$ are both smooth at $(0,0)$ such that the tangent line of $V(f)$ and $V(g)$ is not the $y$ axis. Let ...
2
votes
0answers
186 views

intersection multiplicity and tangents

I haven't been able to find a proof of the following fact, which I have seen mentioned a few times: two non-singular curves have multiplicity intersection greater than 1 at a point P if and only if ...
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votes
0answers
24 views

How do find a projective transformation taking a line to another line

How to find a projective transformation taking line $\{x_{0}+2x_{1}+3x_{2}=0\}$ to $\{x_{0}=0\}$ in $\mathbb{P}^{2}$? I'm thinking trying a few points and solve for the associated matrix. But I can't ...
2
votes
1answer
56 views

Singular curves

How to prove, for example, that there is unique algebraic structure on the curve $\mathbb C P^1 \cup \mathbb CP^1$, where components intersects in 1 point? This is often used in the theory of stable ...
3
votes
2answers
56 views

Poles of abelian differentials

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have ...
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votes
1answer
1k views

How to parametrize the curve of intersection of two surfaces in $\Bbb R^3$?

I have to parametrize the curve of intersection of two surfaces. The surfaces are: $$y^2 = z \text{ and } x + y = 4$$ Could someone please show me how to do this step by step? Thanks.
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2answers
79 views

How to derive a cubic equation $ax^3+bx^2+cx+d =y$ from $x$ and $y$.

Please let me show how to derive a cubic equation form $ax^3+bx^2+cx+d =y$ by using a set of $x$ and $y$ data. Simply the outline of the cubic equation derivation... Thank you
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vote
1answer
126 views

Rational map on smooth projective curve

Let $f:C \rightarrow C'$ be a rational map, here $C$ and $C'$ are smooth projective curves. I cannot understand how is $f$ a morphism? (This is lemma from book by Klaus Hulek) Thanks
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vote
2answers
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Embedding of projective line

Let $C$ be the curve $x^2=yz$ in $P^2$, and $K=K(C)$ be its function field. Is it true that for any polynomial $t$ in $S[x,y,z]$, $k=k(t)$ is function field of $P^1$. Its not clear to me what does ...
2
votes
0answers
73 views

Max Noether's theorem application

I'm trying to solve this problem that I've found on the Internet related to Max Noether's theorem [AF+BG theorem (also known as Max Noether's fundamental theorem)] . It uses the notation of Fulton's ...
0
votes
2answers
48 views

Why does an algebraic curve over an algebraic closed field have smooth points?

Is there an easy way to see this fact? I could try to show that the differential of the defining polynomial cannot vanish at all the zeros. However, I don't see how this could be done. Also there ...
2
votes
1answer
90 views

General surface no lines

I've been studying surfaces recently and I came across the following statement: A general surface $S \subset \mathbb{P}^{3}$ of degree $m \geq4$ contains no lines. Does anyone have any idea how to ...
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0answers
39 views

How do I determine whether the image of a function lies in an algebraic curve?

On p. 2 of the book Differential Equations and Dynamical Systems by Lawrence Perko we define the function $$x(t) = \left(c_1 e^{-t}, c_2 e^{2t}\right)$$ It is then stated that the curve defined by ...
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votes
0answers
74 views

Prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$

I'm meant to prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$. I have a proof for this statement that uses the genus formula to show that such a curve must ...
0
votes
1answer
71 views

B-spline parameterization and derivatives

I have a question regarding the re-parameterisation of a B-spline. Some info: The B-spline is of order 4 (degree 5), hence $C^3$ continuity There is no knot multiplicity The end conditions are not ...
4
votes
1answer
153 views

Frey Curve as a Solution to FLT

I have read in many places that the Frey curve (if it existed) $y^2=x(x-A)(x+B)$ (or equivalently, $y^2=x(x-A)(x-C)$ corresponds to the solutions of $a^n+b^n=c^n$, where $A=a^n/c^n$ and $B=b^n/c^n$. ...
4
votes
0answers
39 views

Lifting extensions of sheaves

Consider a curve $X_0$ over an algebraically closed field $k$. Let $A$ be a local Artinian ring over $k$ and $X$ a scheme over $A$ with $X \otimes_A k = X_0$. For a sheaf $F$ on $X$ we denote by $F_0$ ...
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52 views

Vanishing of sections and special divisors

Let $L$ be a line bundle on a smooth complex projective curve $X$. Suppose we have vector subspaces $$U\subset V\subset H^0(X,L),\,\,\,\textrm{and}\,\,\,\dim\, U\leq k,\,\,\dim\,V=k+1.$$ I wonder if ...
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97 views

The ideal sheaf defining a curve

Consider a smooth projective survace $X$ over $\mathbb C$, i.e. $X$ is a smooth complex projective variety of dimension $2$. Moreover let $C\subset X$ be an irriducibile curve, clearly $C$ can be ...
3
votes
1answer
49 views

Torsion Subgroups and Periodicity

I am trying to piece together elliptic curves in FLT and would greatly appreciate corrections to my summary (or attempts thereof). Mazur's paper "Number Theory as Gadfly" states, "there is a natural ...
2
votes
1answer
78 views

Cup product mapping

$ \newcommand{\OXD}{\mathcal{O}_X(D)} \newcommand{\OXDD}{\mathcal{O}_X(D')} $ Let $X$ be a smooth projective curve over $k=\bar{k}$ an effective $D$ a divisor on $X$. Associated to $D$ we have a line ...
2
votes
1answer
235 views

multiple tangent lines to a plane curve

Suppose that $C=V(F)$, with $F\in k[X,Y]$ ($k$ algebraically closed), is an irreducible plane curve such that $P=(0,0)$ is a nodal singular point (or an ordinary multiple point according to Fulton's ...
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27 views

Miranda's Exercise J Pag. 167

This is the exercise: If $v^{2}=h(u)$ defines a hyperelliptic curve of genus $g$, then $\phi=[1:u:u^{2}:\dots,u^{g-1}]$ defines a degree $2$ map onto a rational normal curve of degree $g-1$ in ...
3
votes
1answer
113 views

Map of smooth curves and its separability degree

I'm interested in a proof of the following fact from Silverman: Arithmetic of Elliptic Curves: Let $\Phi: C_1 \rightarrow C_2$ be a nonconstant map of smooth curves. Then for all but finitely many ...
4
votes
1answer
51 views

Computing a quotient of rings

Let $R=k[x,y]/(y^2-x^2-x^3)$ and $I=(x,y)\cdot R \subset R$. I would like to show that $$ \bigoplus_{i=0}^{\infty} I^i\,/\,I^{i+1} \cong \,k[x,y]\,/\,(x^2-y^2). $$ Could you please help me? Remark: ...
2
votes
1answer
125 views

About Linear Systems on Curves.

Let $C$ be a smooth irreducible (complex) curve of genus $g\geq2$. The gonality of $C$ is defined as the minimum degree of surjective morphisms $C\rightarrow\Bbb{P}^1$. So $C$ has gonality $d$ if it ...
2
votes
1answer
53 views

Is this union of tangent spaces a known object in Algebraic Geometry?

Let $C$ be a family of smooth (all but one curve is smooth, I'll explain later) cubic curves in $\mathbb{P}_{\mathbb{C}}^2$ with the following property: given any two elements $a,b \in C$, the curves ...