problem using ajax url

You should use the WP AJAX APIs rather than reinventing them in your home template, e.g.:

functions.php :

// declare the URL to the file that handles the AJAX request (wp-admin/admin-ajax.php)
wp_localize_script( 'my-ajax-request', 'MyAjax', array( 'ajaxurl' => admin_url( 'admin-ajax.php' ) ) );

add_action('wp_ajax_my_action', 'my_action_callback');
add_action('wp_ajax_nopriv_my_action', 'my_action_callback');

function my_action_callback() {
    $content="something to send back to the browser";

    echo $content;

    die(); // this is required to return a proper result
}

Javascript:

jQuery.post(
    // see tip #1 for how we declare global javascript variables
    MyAjax.ajaxurl,
    {
        // here we declare the parameters to send along with the request
        // this means the following action hooks will be fired:
        // wp_ajax_nopriv_myajax-submit and wp_ajax_myajax-submit
        action : 'myajax-submit',

        // other parameters can be added along with "action"
        postID : MyAjax.postID
    },
    function( response ) {
        alert( response ); // 
    }
);

Using this you can bypass your broken code and rely entirely on standardised APIs

You can read more about it here:

http://codex.wordpress.org/AJAX_in_Plugins

It also applies to themes