Answer: It sounds like your seedlings are "damping off". Here are a few methods you can try, although it won't help with the ones already "gone".
Since the fungus enters the plant at the soil line you can try to make an inhospitable environment for the fungus. When the seeds are planted, but before they germinate, cover the soil with a fine layer of "play sand" (this is sand that has been sterilized; you should be able to find it at a hardware store or lumberyard). This provides a sterile, dry -- and therefore unfavorable to the fungus -- surface at the point it usually enters the stem. Provide good, very good, air circulation (a fan blowing gently in the room--but not directly on plants--is one way), and be sure not to overwater. Another trick is to hold off on fertilizing the seedlings until their second set of leaves appear.
If you use a sterile medium (and it's a good idea to do so), sterilize your pots before planting with a 10 percent bleach solution, and follow the above steps, you should not need to apply a fungicide.
Also, once the plants have germinated, it's best keep them under the lights for about 16 hours per day. Keep the lights close to the tops of the seedlings--just an inch or two above the tips.
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