1
vote
0answers
102 views

Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
1
vote
0answers
18 views

automorphism groups of hyerelliptic curves in positive charactersitic

It appears that the automorphism groups of hyperelliptic curves are at least well studied, if not understood, in the characteristic zero case. I would imagine that most of these results would carry to ...
4
votes
3answers
78 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
5
votes
1answer
50 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
2
votes
1answer
153 views

Elliptic curve as an intersection of quadrics

Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
2
votes
1answer
87 views

Reference request: Construction of $M_{1,0}$

Does anyone know a reference for the construction of the (Artin) stack $M_{1,0}$ and a result about the corresponding coarse moduli space? In Deligne-Mumford they construct $M_{g,0}$ when $g\geq 2$ ...
6
votes
1answer
174 views

How much do I need to learn before I can read about Toric varieties?

I have a copy of the book "Introduction to Toric varieties" by William Fulton, and over the next few months I'd like to make some progress on it. As a first goal, I'd like to be able to read just ...
1
vote
1answer
36 views

Does pull-push by the quotient map of divisors on the symmetric square of an algebraic curve induce multiplication-by-2?

Let $C$ be a smooth projective algebraic curve over a field $k$ of characteristic different from 2, and let $C^2 = C \times C$ be the square of $C$. Let $C^{(2)} = \operatorname{Sym}^2(C)$ be the ...
4
votes
1answer
111 views

Algebraic vs. Analytic curves

I'm familiar with the idea of using algebra to study certain types of plane curves, and my understanding is that there is a whole class of "algebraic curves" that can be studied this way. It would ...
9
votes
2answers
351 views

Elementary proof of the Hurwitz formula

I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective ...
5
votes
1answer
167 views

Relationship between two distinct notions for divisors on curves

I've seen divisors on curves before, a few years ago in a course in algebraic geometry. Now I've come across them again, but they're somewhat more generalized. I was hoping someone could explain the ...
9
votes
2answers
349 views

Good books/expository papers in moduli theory

I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves. I began ...