Timeline for Complement of the Brill Noether locus in $\mathcal{M}_g$
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2016 at 18:25 | comment | added | meh | @user... I don't know of a good description. Surely Jason is far more expert in such things as I. Perhaps one can make a statement about injectivity of the Petri maps as the criteria. My statement, as well as the statement in ACGH (I believe) are about one general curve. | |
| Oct 4, 2016 at 15:22 | comment | added | user52991 | @aginensky, I would like to know what is meant by general in that context (when $W^r_d$ are irreducible). It means such curves for an open subset of the moduli space. Is there a description of this open set or codimension of its complement? | |
| Oct 4, 2016 at 15:13 | comment | added | meh | With regard to user52991's first response to Jason, my recollection is that in ACGH, at the point the comment is made, one also knows that on a general curve, all the $W^r_d$ are smooth. From Fulton-Lazarsfeld we have that the $W^r_d$ are connected. That $W^r_d$ is irreducible follows. | |
| Oct 4, 2016 at 13:08 | comment | added | Jason Starr | The question you asked above is different from the question you asked in your most recent comment. Nonemptiness of $W^r_d(C)$ is different from irreducibility of $W^r_d(C)$. Does that help clear up your confusion? | |
| Oct 4, 2016 at 13:02 | comment | added | user52991 | @Jason Starr. Ah sorry, I don't think I follow your comment. In the corollary to the connectedness theorem, it says for a general curve $C$, if $\rho\geq 1$, then the variety is irreducible. What does general mean in this context? Because $\rho< 1$ is not valid here. | |
| Oct 4, 2016 at 12:56 | comment | added | Jason Starr | No, not all curves are Brill-Noether general. However, you defined $W^r_d(C)$ using an inequality, i.e., $h^0(L) \geq r+1$ rather than $h^0(L)=r+1$. With that definition, whenever $\rho\geq 0$, the set $W^r_d(C)$ is nonempty for all smooth, genus $g$ curves $C$. Saying that $C$ is "Brill-Noether general" usually means that for all $\rho$ with $\rho<0$, $W^r_d(C)$ is empty (some authors would add the condition that the spaces $G^r_d(C)$ are all smooth). | |
| Oct 4, 2016 at 12:12 | comment | added | user52991 | @Jason Starr. Thank you. The purpose of my question is as follows. In ACGH, it says that as a corollary of the Fulton Lazarsfeld connectedness theorem, for a general curve $C$, if $\rho\geq 1$, then $W^r_d(C)$ is irreducible. I thought 'general' here meant Brill Noether general. But all curves in the moduli space are Brill Noether general? | |
| Oct 4, 2016 at 11:56 | comment | added | Jason Starr | If $\rho$ is nonnegative, then the set that you call $\mathcal{W}^r_d$ equals all of $\mathcal{M}_g$. When $\rho$ equals $-1$, $\mathcal{W}^r_d$ is a divisor. This is part of the proof that $\overline{\mathcal{M}}_g$ is of general type for $g$ large (Harris, Mumford, Eisenbud). | |
| Oct 4, 2016 at 11:53 | history | asked | user52991 | CC BY-SA 3.0 |