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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34 views

Rational locus of a function defined on $x^2+x^3=y^2$

We have a curve $X$ on $\mathbb{A}^2$ given by $y^2=x^2+x^3$. Consider the rational function $f$ on $X$ which maps $(x,y)\in X$ to $\frac{y}{x}$. There is a nice geometric interpretation of $f$: if we ...
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47 views

When is $\mathcal{H}om$ functor exact in the category of presheaves

Let $C$ be a projective $\mathbb{C}$-scheme of pure dimension $1$. Suppose that $C$ is local complete intersection in $\mathbb{P}^3$. Let $C_1$ be an irreducible component of $C$, also of pure ...
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1answer
53 views

Alternative models of a hyperelliptic curve

I am studying some particular examples of hyperelliptic curves and their automorphism groups (from this paper, if it is of interest). It is mentioned in the paper to understand the automorphism of ...
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0answers
31 views

Smooth conics in linear subspaces of $S^2U^*$

Let $U$ be some $3$-dimensional vector space over some field $\mathbb{k}$. It is possible to consider the projective space $\mathbb{P}(S^2U^*)$ as a space of conics on the projective plane ...
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1answer
161 views

Projective equivalence

Definition Two projective plane curves $F$ and $G$ are projectively equivalent if there is a $\varphi_A\in PGL_2(k)$ such that $F(x,y,z)=G(a_{00}x+a_{01}y+a_{02}z,a_{10}x+a_{11}y+a_{12}z, ...
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153 views

Normalization of a curve

This is a point in an exercise given during my Commutative Algebra course. Let $k$ be an algebraically closed field with characteristic different from $2$. Let $R=\frac{k[x,y]}{(y^{2}-f(x))}$, where ...
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29 views

Questions on linear subspace of a projective space

I am a bit confused by the definition of the linear subspace of a projective space. It says in a book "Algebraic Geometry: A first course" by Joe Harris on page 5 that An inclusion of subspace ...
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0answers
142 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
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162 views

Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

According to this article, The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. But I ...
4
votes
2answers
83 views

Two conics have exactly one intersection point

We have two conics $Q_1,Q_2$ on $\mathbb{P}_2$ over some algebraically closed field. Also $Q_1$ and $Q_2$ are supposed to be smooth. I've just discovered Bezout's theorem, which states that two ...
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5answers
122 views

Determine cubic function from 2 roots and a maximum.

If I am trying to find a cubic function with 3 real roots, and I know two of them, and one local maximum, is it possible? Assuming my roots are $0.05$, $0.95$ and $u$, and my local maximum is $(i, ...
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0answers
39 views

computing the divisor of a differential

I have some trouble to computer the divisor of a differential in the subject of algebraic curve. Any feedback is greatly appreciated.Thank you.
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0answers
64 views

$V(f)$ is irreducible iff $f=g^k$, $g$ irreducible

I'm trying to prove this theorem $V(f)$ is irreducible iff $f=g^k$, $g$ irreducible. To prove the converse, we have $V(f)=V(g^k)=V(g)$, since $g$ is irreducible $V(g)$ is irreducible, then ...
4
votes
1answer
85 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
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1answer
98 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
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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 ...
4
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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
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1answer
68 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
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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
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0answers
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
74 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 ...
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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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59 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
110 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
61 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
84 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
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65 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
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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
193 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 ...
5
votes
1answer
191 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
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0answers
71 views

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

We have, $$x^{12} - a x^{11} - 33x^8 + 22a x^7 - 11a^2 x^6 + 363x^4 - 121a x^3 + 121a^2x^2 - 23a^3x - 11^3 + a^4=0$$ $$x^{12} - a x^{11} - 11a x^9 - 44a x^7 - 88a x^5 - 88a x^3 - 32a x - a^2=0$$ ...
2
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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 ...
1
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0answers
44 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
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0answers
190 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
25 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 ...
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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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1answer
2k 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
83 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
1
vote
1answer
129 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
79 views

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
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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 ...
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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 ...
0
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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 ...