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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1answer
64 views

Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
4
votes
1answer
55 views

Smoothness of the Picard group of a smooth curve

Let $X$ be a smooth projective curve over $k=\bar{k}$ and denote its Picard group by $\operatorname{Pic}(X)$, with the usual scheme structure coming from the representability of the relative Picard ...
1
vote
1answer
111 views

Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
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votes
0answers
13 views

Easy question about coordinate rings

Let $C=V(x^2+y^2-1)$ be an affine algebraic curve. In an online course the professor said $\varphi=\frac{x-1}{y}\in A(C)$, but he didn't explain why. I would like to know how he gets this function ...
4
votes
1answer
65 views

inconsistency of the Plücker's formula

I'm a beginner in algebraic curves and as an exercise I'm playing with the Plücker's formula. I'm finding some inconsistency in these formulas and I would like to know where I'm wrong. We know the ...
2
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0answers
27 views

Bitangents corresponds to nodal points in the dual space

I'm beginning to study algebraic curves and I couldn't prove if we have $L$ a line bitangent to $F$, i.e, there are points $p_1, p_2\in F$, such that $L=T_{P_1}F=T_{P_2}F$, then $P_L\in F^\vee$ is a ...
4
votes
1answer
96 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
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votes
1answer
48 views

Why the inverse of the matrix in this definition

We have this definition: I didn't understand why in this definition we just define $F^A$ to be $$F^A(X_0:X_1:X_2)=F\bigg((X_0:X_1:X_2)A\bigg)$$ $F^A$ is not the composition $F\circ c$? Why ...
3
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0answers
97 views

$XY^4+YZ^4+XZ^4$ has no singular points

In the question 5.1 in the Fulton's algebraic curves book he asked to find the multiple points of $$F=XY^4+YZ^4+XZ^4$$ Calculating the partial derivatives, we have: $\frac{\partial F}{\partial ...
2
votes
2answers
27 views

Help in this very basic example in algebraic curves

I'm trying to understand this example: I didn't understand why the second factor describes a point of intersection $q$, since the second factor doesn't vanish at $q$. Anyone can clarifies this for ...
3
votes
1answer
74 views

The intersection numbers in Fermat curve

I'm a beginner in this subject and I think this "easy" exercise could help me to have more practice in basic algebraic curves. Let $F=X^{p+1}+Y^{p+1}+Z^{p+1}$ be a Fermat curve in the field $k$, with ...
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vote
0answers
37 views

Vector bundle on a transversal intersection

Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such ...
2
votes
1answer
67 views

existence of a line on a cubic surface

I'm trying to understand the proof in Miles Reid's book: Undergraduate Algebraic Geometry, that: "there exists at least one line l on S", where S is a non-singular cubic surface. In the book the proof ...
3
votes
1answer
72 views

Degree of the dual curve to $XY^2 - Z^3$

I have a question about the dual curve to the curve $C$ cut out by the equation $F(X,Y,Z) = XY^2 - Z^3 = 0$ in $\mathbb{P}^2$. (Assume that everything is over an algebraically closed field of ...
2
votes
1answer
31 views

Singular varieties

Let $y^2=x^3+ax+b$ and V be its affine variety. V is singular iff $y^2-x^3-ax-b$, 2y, and $3x^2+a$ have a common zero iff $x^3+ax+b$ and $3x^2+a$ have a common zero iff $x^3+ax+b$ has a multiple root. ...
0
votes
1answer
35 views

Bijection between the projective plane and its dual

I didn't understand why we can't identify the projective plane with its dual. Let's take for example a line $L=aX+bY+cZ$ with $(a,b,c)\neq (0,0,0)$ in the projective plane $\mathbb P^2$. The dual ...
2
votes
1answer
41 views

Intersection number of the tangent at the Inflexion point of $y=x^3$

We know that the intersection number of this curve $f=y-x^3$ and its tangent at the origin is $3$. I'm trying to use this method described in the Fulton's book: Following this definition we have ...
1
vote
1answer
78 views

through two points passes an unique line and generalizations

I would like to prove for each integer $d\ge 1$, there are $\frac{d(d+3)}{2}$ points in the plane for with passes exactly a curve of degree $d$. For $d=1$, according to the statement there are 2 ...
4
votes
0answers
84 views

no morphism between curves of different genus

Let $C_1,C_2$ be two smooth irreducible curves of genus 4,3 resp. Prove there is no morphism $\phi: C_1 \to C_2$. Well, the tool of treating genera is Hurwitz's theorem, which says here that ...
2
votes
2answers
133 views

Selecting a book for a reading course in Algebraic Geometry

I'm not sure if this is an appropriate question in this forum, but here is the situation. I must begin by saying that I know basically nothing about Algebraic Geometry, but this semester I will be ...
5
votes
1answer
59 views

Finding all morphism from a variety to itself

Let $$C:=X^2+Y^2-Z^2$$ be a projective variety in $\Bbb P^2$. What are all the morphisms $C\to C$ ? More generally, how does one find all morphisms from a given variety to itself? ...
3
votes
2answers
106 views

$Im(\phi)$ is closed subset of $\mathbb{A}^2$

let $\alpha(t)$ and $\beta(t)$ $\in$ $K[t]$ , $\phi(t)=(\alpha(t),\beta(t))$ is a morphism from $\mathbb{A}^1$ to $\mathbb{A}^2$ show that $Im(\phi)$ is closed subset of $\mathbb{A}^2$. it seems ...
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votes
0answers
124 views

Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto ...
2
votes
0answers
113 views

Parametrization of the cuspidal cubic

I didn't understand why the method works fine to find the parametrization of the cuspidal curve: I didn't understand why finding these intersections points will give me the whole curve. thanks
3
votes
2answers
99 views

Parametrization of the line in the projective space

Let $L=aX+bY+cZ$ be a line in the projective space, the book I'm using states that every such line has the following parametrization: $$\varphi:\mathbb P^1\to L, \ (t:s)\mapsto ...
0
votes
0answers
50 views

Construction of rational function on projective curve

Let $k(X)$ be the rational function field of $X,$ where $k$ is an algebraically closed field and $X$ is a nonsingular projective curve. Let $U \subseteq X$ be open and let { $U_{i}$ }, $i\in I,$ a ...
8
votes
1answer
89 views

Inhomogeneous polynomial and points at infinity

Let $f=X^2-Y$ be a polynomial in $k[X,Y]$, so $V(Z)$ is a parabola: $V(f)$: According to Bézout theorem the $y$-axis has to intersect the parabola two times. We know the y-axis meets the ...
3
votes
1answer
82 views

Example of a curve of genus $4$

I'd like to put my hands on some polynomial defining a curve of genus $4$, living in the plane or in the 3D space. Do you know about any? Is there any procedure to build one? The best would be one ...
0
votes
1answer
35 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 ...
0
votes
0answers
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 ...
3
votes
1answer
56 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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votes
0answers
32 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 ...
2
votes
1answer
179 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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votes
0answers
176 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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0answers
31 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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vote
0answers
163 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 ...
8
votes
1answer
169 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
85 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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vote
5answers
130 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
44 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
65 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
88 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
101 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
116 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
votes
0answers
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
71 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
49 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
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
76 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
39 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 ...