Answer: Blossom end rot is a common problem and sounds like what you describe---EXCEPT that the rot is on the bottom, or blossom end, of the tomato, not the stem end. Just want to mention this in case you could be mistaken about the location of the rot. This disorder is caused by uneven water availability, and perhaps calcium deficiency, not a fungus.
If the problem is indeed on the stem end, it could be any of several diseases, though it sounds most like anthracnose. Anthracnose fungal spores overwinter on plant debris. They're released during wet weather and spread by wind and splashing water onto tomato plants. Although the disease can infect tomato leaves, it's primarily a problem on the fruits. The disease isn't visible on green fruits, but once they begin to ripen, sunken, 1/2 inch diameter black spots develop. Secondary rot organisms follow the initial infection and the tomatoes and, unfortunately, can't be saved. Anthracnose can occur all season long but is mostly a problem during wet weather and when plants have a severe early blight infection. To control the disease, try to prevent the spores from landing on developing fruit by planting tomatoes in black plastic or mulching with a thick layer of straw. Other prevention measures such as thoroughly cleaning up any plant debris in fall and rotating tomatoes to another part of the garden will help. You may need to resort to fungicide sprays--there is new, low-toxicity fungicide called "Soap Shield" that is labeled for anthracnose; it is available from Gardens Alive (ph# 812-537-5108, e-mail firstname.lastname@example.org)
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