Is it possible to fine-tune email alerts?

I'd like to get email when low on time, or challenged, but not when I am just to move...

That be good to have... I'm tired of clearing my inbox of "It's your turn to move in your game against PersonName" messages, but I do want to know when I'm challenged or low on time.

You can regulate how much email you get at any time.Click here to adjust your alert and notification settings.

You can also use this one  to set game alerts

Game Alert Preferences

kohai, I visited this screen before asking the question. There is single checkbox "online chess" for my move, low on time, game challenge, etc.

The alert sent from the "alert & email" setting will show up as a pop up reminder on your pages here, but game alert preferences on the "online settings" page should only be sending you email reminders when you're not logged into the site

Very confusing. But still, on the latter screen one can also see:

[x] alert me when it is my move, I am running out of time, or I receive a challenge

I'd like to be alerted when running out of time, or I receive a challenge, but not when it is my move...

i'd like to be alerted that it's my move even if I'm logged in.  Is that possible?

Yes. Use this link then check the box marked  "Also send alerts to my xxxxxx email address

Use the drop down box there to mark whether you want the alerts to be sent to your @chess.com email address or your default external email address