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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6
votes
2answers
97 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
2
votes
1answer
54 views

Does $\,f_* \mathcal{O}_{X_T} \cong \mathcal{O}_{T}$ hold in this situation?

Let $X$ be a scheme over $S$ and consider the following hypothesis : \begin{cases} \; (1) \quad f:X\to S \text{ is quasi-compact and quasi-separated } \\\\ \; (2) \quad f:X\to S \text{ admits a ...
1
vote
0answers
39 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
1
vote
0answers
28 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
6
votes
1answer
95 views

Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
3
votes
1answer
36 views

Curve with acnodes over closed fields?

From Wikipedia: An acnode is an isolated point not on a curve, but whose coordinates satisfy the equation of the curve. The term "isolated point" or "hermit point" is an equivalent term. I was ...
3
votes
0answers
72 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
3
votes
4answers
172 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
6
votes
1answer
153 views

Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
4
votes
0answers
134 views

Simple proof of an equality on curves, with or without local fields

Is there a simple way to see that, given a nonsingular curve $X$ and a finite morphism $f\colon X\to Y,$ then $$2\cdot c_1(f_*\mathcal O_X)=-f_*R_f,$$ where $R_f$ is the ramification divisor of $f?$ I ...
3
votes
1answer
72 views

Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...
14
votes
0answers
184 views

Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
4
votes
2answers
61 views

Given $\omega_i \in \Omega_X(U_i)$ can I find $f\in {\cal O}_X(\cap U_i)$ so that $df = \omega_1 - \omega_2$

As per this question: Duality in algebraic de Rham cohomology I am trying to show that the map $H^1(X,\Omega_X) \rightarrow H^2_{\text dR}(X/k)$, where $X$ is a projective algebraic curve over an ...
3
votes
1answer
40 views

Help in this question in Fulton's algebraic curves

I'm trying to solve this question: In item (a) I used the fact $O_a(V)$ is a Noetherian local ring and the only maximal ideal is $(x-a)$. First note that the non-units of $O_a(V)$ are the elements ...
4
votes
1answer
56 views

What is this cycle on the Jacobian of a curve?

Let $C$ be a curve and $J$ it's Jacobian. There is the standard Abel-Jacobi map $a:C\rightarrow J$ which is given by $Q\rightarrow Q-P$ for some fixed point $P$ (here I am regarding $J$ as the degree ...
0
votes
1answer
19 views

which regression is better

suppose that we have two input vector and the variables in each vectors are independent and uncorrelated from each other,just only there is relationship between two vector,but not itself in ...
1
vote
1answer
65 views

Why is this is an equivalence relation?

Fulton makes the following definitions: After he defines an equivalence relation: The definitions he made seems very obscure to me and if anyone could show why this relation is an equivalence ...
5
votes
1answer
79 views

What is normal crossing?

I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with ...
3
votes
1answer
62 views

Why there is a minimal element of this set

I'm trying to understand this proof: I know intuitively, but Why formally there is such a minimal element? I need help Thanks
2
votes
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
53 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
110 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 ...
0
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
64 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
votes
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
95 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 ...
0
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
votes
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
73 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 ...
1
vote
0answers
36 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
66 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
70 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
30 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
33 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
40 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
70 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
82 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
127 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
58 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
104 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 ...
4
votes
0answers
122 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
96 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
95 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
88 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
79 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
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 ...
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
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 ...