1
vote
0answers
217 views

A question on partial fraction decompositions

This question concerns a mapping from the poles of a rational function to its partial fraction decomposition coefficients. We assume that the rational function is the inverse of a polynomial of degree ...
0
votes
0answers
167 views

sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...
1
vote
0answers
149 views

Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
12
votes
2answers
734 views

Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
8
votes
1answer
323 views

Given a curve, under which condition is the set of gonal morphisms finite

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...
2
votes
1answer
268 views

resolution of singularities and a projection formula

Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities. Let $f$ be a rational function on $Y$. Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as ...
1
vote
1answer
312 views

Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar

Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$. Q1. ...
1
vote
1answer
205 views

Rationality of quadric fibrations

Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following? $Q$ is rational if and only if $Q \to S$ has a ...
12
votes
0answers
535 views

Pencils with many completely decomposable fibers

Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree in $\mathbb C^{n+1})$. The fiber over $(\lambda:\mu) \in ...
7
votes
5answers
1k views

Rational maps with all critical points fixed

What can be said about rational self-maps of $\mathbb P^1$ for which all critical points are also fixed points ? If all but one of the fixed points are critical, there is a characterization in ...