How To Write Into A Cross Subdomain Cookie

I manage a site with a normal address like: www.mysite.com
I'm using a cookie to store nicks of my users, with a code like this:
----
Response.Cookies("mycookie").Expires = #January 1, 2030#
Response.Cookies("mycookie").Path = ""
Response.Cookies("mycookie").Secure = FALSE
Response.Cookies("mycookie")("nick") = nick
----

and I'm deleting it, if a user wants so, with:
----
Response.Cookies("mycookie").Expires = #January 1, 2003#
Response.Cookies("mycookie").Path = ""
Response.Cookies("mycookie").Secure = FALSE
Response.Cookies("mycookie")("nick") = ""
Response.Cookies("mycookie") = ""
----
(I know, there's too much stuff in the delete procedure!)

Now I'm going to open some new sections of my site, with subdomain
adresses, like search.mysite.com, shop.mysite.com... and I need that
cookie authentication for the main (www) address is working also in
these subdomains.

I red on the Internet that with the DOMAIN property, I can set my
cookie to make it work for every subdomain, with something like:
----
Response.cookies("mycookie").domain = ".mysite.com"
----

.... and it works!
My problem is when I try to DELETE it: after I set a cookie with
..domain property, it seems to become "read only": any my attempt to
delete (or modify) it seems to be ignored.

Results are the same when I try to write it from the subdomain that
created it (www), or when I try from the "chief" address (mysite.com,
without any subdomain).

Where's my mistake?