Timeline for Reference for map $\operatorname{Hom}^d(C,\mathbb{P}^1) \to \operatorname{Sym}^d(C)$

10 events

when toggle format	what		by	license	comment
Nov 22, 2018 at 16:59	history	edited	R. van Dobben de Bruyn		edited tags
Dec 25, 2017 at 23:36	vote	accept	Krijn
Jun 23, 2017 at 3:14	answer	added	R. van Dobben de Bruyn		timeline score: 3
Jun 22, 2017 at 22:26	history	edited	Krijn	CC BY-SA 3.0	deleted 241 characters in body
Jun 22, 2017 at 22:23	comment	added	Krijn		@R.vanDobbendeBruyn As I am mostly interested in large $d$, we may assume that $d > 2g$ so that we can view any point $D \in \operatorname{Sym}^d(C)$ as an effective divisor and so $\phi$ is surjective. I'll edit this in.
Jun 14, 2017 at 13:49	comment	added	R. van Dobben de Bruyn		For an example of non-flatness, look at $C$ hyperelliptic of genus $g \geq 2$ with $d = 2$. Then the degree $2$ map $C \to \mathbb P^1$ is unique up to $\operatorname{PGL}_2$. Thus, the image of your map $\phi$ is the locus given by divisors $P+Q$ linearly equivalent to the unique $g^1_2$ (equivalently, $P+Q$ is a fibre of this unique degree $2$ map). Most pairs $(P,Q)$ do not satisfy this, so the image of $\phi$ is a strict closed subvariety. Hence, $\phi$ cannot be flat.
Jun 14, 2017 at 10:53	history	edited	YCor		edited tags
Jun 14, 2017 at 10:34	history	edited	Krijn	CC BY-SA 3.0	Added information
Jun 8, 2017 at 13:01	review	First posts
Jun 8, 2017 at 13:05
Jun 8, 2017 at 12:57	history	asked	Krijn	CC BY-SA 3.0