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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27
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367 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 ...
8
votes
0answers
132 views

Tropical-like redefinitions of addition and multiplication?

I've been impressed at the rich structure of tropical mathematics, all consequences of the seemingly mundane starting point of replacing classical $a+b$ by $\min(a,b)$ (or max), and replacing ...
7
votes
0answers
130 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
7
votes
0answers
222 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
7
votes
0answers
475 views

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 ...
6
votes
0answers
96 views

A Regular Map Has Finitely Many Ramification Points

Let $C,D$ be nonsingular projective curves, $f \colon C \to D$ nonconstant, $K = k(C), L = k(D)$, $d = \text{deg }f = [K:L]$, and of course $k$ algebraically closed. Furthermore let's suppose that ...
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
0answers
89 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 ...
5
votes
0answers
122 views

Line bundles over a curve

I keep seeing statements like that and I don't know how they might be established: Let $k \subset \bar{k}$ be fields, $C \rightarrow Spec(\bar{k})$ be a curve and $\mathscr{L}$ a line bundle on $C$. ...
5
votes
0answers
78 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
5
votes
0answers
371 views

An argument on page 62 of Griffith's book, “Introduction to Algebraic Curves”

I am a bit confused about some of the things that Griffiths says on page 62 of his book, Introduction to Algebraic Curves. I am not sure how I can reproduce the text here. I can see that GoogleBooks ...
5
votes
0answers
329 views

The Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = ...
4
votes
0answers
65 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
4
votes
0answers
54 views

On the bidegree of a curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence. I'm ...
4
votes
0answers
78 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 ...
4
votes
0answers
135 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 ...
4
votes
0answers
87 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 ...
4
votes
0answers
127 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 ...
4
votes
0answers
43 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 ...
4
votes
0answers
65 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
0answers
40 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 $F_0$ ...
4
votes
0answers
55 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 ...
4
votes
0answers
234 views

Cardinality of the Fiber of a Finite Morphism Vs. Degree (via Vakil)

I would like to show the following (note: it is not an assigned problem, so it may be false) (EDIT: Indeed it is false, see end of post): Suppose $f:X \rightarrow Y$ is a finite, surjective morphism ...
4
votes
0answers
72 views

Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
4
votes
0answers
110 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
4
votes
0answers
138 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
4
votes
0answers
67 views

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}, ...
4
votes
0answers
231 views

intersection multiplicity and partial derivatives of algebraic curves

this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes: Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ...
3
votes
0answers
33 views

Number of rational points on a curve and genus of a curve

I've just started with algebraic geometry, so i apologize in advance if my question is too easy to show. Given is a curve $\Gamma $ in $\mathbb{P}^{2}(\mathbb{F_{q^{m}}})$ defined by ...
3
votes
0answers
49 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
3
votes
0answers
37 views

When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
3
votes
0answers
62 views

How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space?

I have the equation of two 5th degree polynomials which they don`t intersect with each other .Each curve is made of 100 points and these two curves looks similar but there are small differences .I am ...
3
votes
0answers
70 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
3
votes
0answers
63 views

Are $k$ points on a smooth algebraic plane curve ever in general position?

Let $C$ be a smooth plane curve of degree $d$ and genus $g=\frac{(d-1)(d-2)}{2}$. Let us choose $k\leq g+3d-1$ points on $C$. Is it true that the dimension of the space of plane curves of degree $d$ ...
3
votes
0answers
44 views

Automorphisms of rational curves

Let $X$ be a non-empty open subscheme of $\mathbb P^1_{\mathbb C}$. What is the automorphism group of $X$ in terms of PGL$_n(\mathbb C)$ and the points on the boundary?
3
votes
0answers
53 views

Inequality involving multiplicities of points introduced via Quadratic Transformations of a Plane Curve

I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF). The question is: Let $F=F_1, ...
3
votes
0answers
127 views

Computing the genus in positive characteristic

That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to ...
3
votes
0answers
99 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 ...
3
votes
0answers
66 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 ...
3
votes
0answers
71 views

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

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

Linear system of degree $d$ curves passing $m$ times through $P$ in the blow-up at $P$.

Given a point $P$ in $\mathbb{P}^2$ and a natural number $m$ we consider the linear system $\mathcal{L}$ of curves of degree $d$ passing $m$ times through $P$. If $H$ is the line class of the plane, ...
3
votes
0answers
79 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
3
votes
0answers
54 views

Spectrum theorem for p-adic matrix analysis

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\ 2: ...
3
votes
0answers
62 views

Is the number of automorphisms of a hyperelliptic curve bounded

Certainly, if we fix the genus $g$ of a curve $X$, we have $\# $Aut$(X) \leq 84(g-1)$. Let $X$ be a hyperelliptic curve. Is there a bound on $\#$Aut$(X)$? (Note that I do not want to fix the genus!) ...
3
votes
0answers
53 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
3
votes
0answers
33 views

Methods to prove that points do not lie on an algebraic plane curve

I have an infinite sequence of points in an affine plane and I want to show that these points do not lie on any algebraic plane curve. Are there any standard methods for doing this?
3
votes
0answers
56 views

Algebraic Curves similar to Hyper-Elliptic Curves

Throughout, $F_q$ will denote a finite field of $q$ elements with characteristic $p \neq 2$. It is well-known that the equation $y^2 = f(x)$ (for square-free $f \in F_q[X]$) defines an hyper-elliptic ...
3
votes
0answers
86 views

Is this a map $X \to \text{Sym}^n Y$?

Let $X$ and $Y$ be smooth projective curves over an algebraically closed field $k$ and let $D$ be a divisor on $X \times Y$ meeting every vertical fiber ($X$ is the horizontal axis). Let's assume that ...
3
votes
0answers
63 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
3
votes
0answers
168 views

Minimal resolution of singularities of Fermat curve

Fix a prime number $p$. Let $F$ be the geometrically connected Fermat curve of exponent $p$ over a number field $K$, i.e., $F$ is given by the equation $x^p+y^p = z^p$. Consider the same equation ...