Here is the main finding from a working paper by Parsons, Sulaeman, Yates and Hamermesh:
What are the main results of the study?
There are three. First, umpires are more likely to call strikes for pitchers who share their race/ethnicity. The second result is an extension of the first: Umpires are more likely to express a preference for their own race/ethnicity only when their behavior is less closely scrutinized: 1) in parks where QuesTec (a computerized system set up to monitor and review an umpire’s ball and strike calls) is not installed, 2) in poorly attended games, and 3) on pitches where the umpire’s call cannot determine the outcome of the at-bat. Finally, game outcomes are influenced by the race/ethnicity match between starting pitchers and home-plate umpires. Home teams are more (less) likely to win a game when their starting pitcher and home plate umpire have the same (a different) race/ethnicity.
Science Daily reports on the paper here. Dan Hamermesh sent me the paper just before I went on vacation. I haven't read the paper carefully, but it appears to be well done. The finding that "favoring your own type" varies with monitoring and game conditions -- i.e. that discrimination is essentially price-sensitive (umps do it when it is cheap) -- is especially interesting. You can get directly from Dan's web page; here is the direct link to the paper (pdf).
Update: Here's an MSNBC story on the paper, with commentary from Dan Hamermesh. A sample: "I expect that [MLB] will not be very happy about this, but the fact that with a little bit of effort this kind of behavior can be altered, that's very gratifying. I wish with society as a whole we could reduce the impact of discrimination as easily as it could be done in baseball."