* Gov. Edmund G. (Jerry) Brown Jr., who has been running hard for the Democratic Party nomination to the US Senate since early last fall, made it official March 10 with a formal announcement in Los Angeles. Also in Los Angeles the previous day, novelist Gore Vidal became the sixth entrant in the Democratic race. Although the governor is considered vulnerable in the November general elections, neither Mr. Vidal nor any of the other candidates is expected to outpoll Brown in the June 8 primary.
* Governor Brown and legislative leaders have come up with a plan to balance the current (fiscal 1982) state budget with a number of measures including tax collection speedups and cuts in some state programs designed to eliminate the projected $5 million deficit.
* Proposition 13, the referendum measure that made Howard Jarvis a national symbol of tax revolt, will undergo some revision if backers of a ''split roll initiative'' on next November's ballot are successful. Roland Vincent, who directed the campaign for Prop 13, is behind the proposed change. It would cut property taxes for people who bought homes after 1975 by 50 percent but raise taxes on business and commercial property. The net revenue gain for local governments, says Mr. Vincent, would be some $1.6 billion.
Taxpayers for California, the group backing the petition drive, must collect 553,000 voter signatures by June 14 to get their measure on the November ballot.