This looks to me like a Jquery issue but anyhow... you need to use the Url object to generate a relative Url. In the preview this had problems in that it didn't pick up the ajax context, but it does now.
One way would be to put something like this into you script:
var myUrl = '<%= Url.Action("GetDetails","Home"); %>';
$.ajax({
type: "POST",
url: myUrl });