User klim puhov - MathOverflow

Upper bound for the order of the group of automorphisms of Riemann surfaces of genus 2

2013-02-17T22:57:32Z

By the Hurwitz's automorphisms theorem there is an upper bound $|\text{Aut}(C)|\leq 84(g-1)$ for all Riemann surfaces $C$ with $g(C)\geq 2$, but it is not sharp if $g=2$. What is the sharp upper bound for genus $2$ Riemann surfaces? Also, I'm asking for the simple proof of the fact that there is no genus $2$ Riemann surfaces with $|Aut(C)|=84$.

The description of Hurwitz groups

2013-02-17T14:12:36Z

Let $G$ be a Hurwitz group, i.e the automorphism group of some Hurwitz surface $C$. Then Hurwitz's automorphisms theorem shows that the quotient map of $C$ by $G$ has ramification points of indexes $2$, $3$ and $7$. My question is how to deduce that $G$ is generated by elements $x$ and $y$ satisfying $x^2=y^3=(xy)^7=1$?

On the group actions on Hurwitz surfaces

2013-02-16T19:59:30Z

Let $C$ be a Hurwitz surface, $G=\text{Aut}(C)$ and $N$ is a proper normal subgroup of $G$. Is there a simple argument (without using of classification theorems) for the fact that $N$ acts on $C$ freely?

I found this fact here see Section 3.

On the construction of the varieties parametrizing special linear series on a curve

2013-02-11T23:53:49Z

Fix an algebraic curve $C$ of genus $g$, and positive integers $d, r$. The variety $W^r_d$ parametrizes complete linear series of degree $d$ and dimension at least $r$ on $C$ and the variety $G^r_d$ parametrizes linear series of degree $d$ and dimension $r$ on $C$. I am trying to understand, what are the scheme structures on $W^r_d$ and $G^r_d$ following the book [A,C,G,H] of Arbarello, Cornalba, Griffiths, Harris (see Chapter 4, $\S 3$).

Fix a Poincare line bundle $\mathcal{L}$ of degree $d$ for $C$ and an effective divisor $E$ on $C$ with $m:=\deg{E}\geq 2g-d-1$, and let $\Gamma=E\times \text{Pic}^d(C)$ the divisor on $C\times \text{Pic}^d(C)$. Denote by $\nu: C\times \text{Pic}^d(C) \to \text{Pic}^d(C)$ the projection.

My first question is why $R^1\nu_{*}\mathcal{L}(\Gamma)=0$, $\nu_{*}\mathcal{L}(\Gamma)$ is locally free of rank $d+m-g+1$ and $\nu_{*}(\mathcal{L}(\Gamma)/\mathcal{L})$ is locally free of rank $m$? In [A,C,G,H] is written that it is a consequence of the base change in cohomology, but I can't figure it out.

Taking the direct image of the short exact sequence $$0\to \mathcal{L}\to\mathcal{L}(\Gamma)\to \mathcal{L}(\Gamma)/\mathcal{L}\to 0$$ on $C\times \text{Pic}^d(C)$ we obtain the long exact sequence $$0\to \nu_{*}\mathcal{L}\to \nu_{*}\mathcal{L}(\Gamma)\stackrel{\gamma}\to \nu_{*}(\mathcal{L}(\Gamma)/\mathcal{L})\to R^1\nu_{*}\mathcal{L}\to 0.$$ Then $W^r_d$ is $(m+d-g-r)$th determinantal variety attached to $\gamma$ and $G^r_d$ is the canonical blow-up of $W^r_d$. I read the Chapter 2 of [A,C,G,H] and understood the constructions of $W^r_d$ and $G^r_d$, but I didn't find there, why $G^r_d$ is a blow-up of $W^r_d$ (with the centre $W^{r+1}_d$ I guess?). So, it is exactly my second question.

Thanks.

Why the Abel-Jacoby map is algebraic morphism?

2013-02-11T20:55:21Z

The Abel-Jacobi map from the algebraic curve $C$ to its Jacobian $J(C)$ is given analitically by $$p\to \left( \ldots, \int^{p}_{p_0} \omega_i,\ldots\right),$$ where $p_0$ is some point on $C$ and $\omega_i$ form a basis of $H^0(C,K)$. Why it is an algebraic morphism?

Total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}^n$

2010-11-06T23:13:46Z

It is well known that total space of the tautological line bundle $\mathcal{O}(-1)$ over projective space $\mathbb{P}^n$ is closed subvariety of $\mathbb{P}^n\times\mathbb{A}^{n+1}$. My question is how to realize total space of $\mathcal{O}(1)$ over $\mathbb{P}^n$ in such manner, i.e. I need an embedding of $Tot(\mathcal{O}(1))$ in simple variety and defining equations. Thanks.

Comment by Klim Puhov

2013-02-17T14:34:01Z

@Peter: Ok, maybe "classification" is not a right word. So, you use the description of finite branched Galois covers of $P^1$. Where is the proof can be found?

Comment by Klim Puhov

2013-02-17T11:18:01Z

@Peter: Your answer is exactly what I called "using of classification theorems". So, unfortunately, it is not helpfull for me. Using Riemann-Hurwitz genus formula, together with the proof of the Hurwitz bound, I can show that the quotient map by the $G$-action have ramification points of indexes $2$, $3$ and $7$, but I can't figure out that $G$ have the description that you give in your answer.

Comment by Klim Puhov

2013-02-17T10:26:26Z

Could you explain, please, why $G$ is generated by a,b,c with relatively prime orders 2,3,7 and abc=1? I can't find the proof in literature.

Comment by Klim Puhov

2013-02-17T10:07:06Z

Could you provide an example? I found this fact here heldermann-verlag.de/gcc/gcc02/gcc028.pdf in Section 3.

Comment by Klim Puhov

2013-02-17T09:36:10Z

No. The quotient map by the $G$-action have ramification points of indexes $2$, $3$ and $7$ (see [wiki][1]). They have non-trivial stabilizers. [1]: en.wikipedia.org/wiki/…

Comment by Klim Puhov

2013-02-12T02:15:07Z

1. Thanks a lot. 2. I still don't understand. Is the "canonical blowup" $\widetilde{X}_k(\phi)$ of a determinantal variety $X_k(\phi)$, as it defined in chapter 2, a blowup of some close subvariety in $X_k(\phi)$ or it is only the name for the construction of incidence correspondence?

Comment by Klim Puhov

2010-11-08T14:23:34Z

Thank you, Tony. Maybe you know such simple description of $Tot(O(n))$ for $n>1$ also?

Comment by Klim Puhov

2010-11-07T20:35:32Z

Yes, I know about such description, but for my purposes the above one is preferable.

Comment by Klim Puhov

2010-11-07T20:22:34Z

Thanks for detailed answer, Georges!