What is the new Google Author markup, and how can I use it to build up a reputation as an online writer?
How could it possibly work? Using self-declared information has been tried before and of course led to spamming and other problems.
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The author mark-up is relatively simple.
Written by <a rel="author" href="http://www.example.com/peter-mortensen">Peter Mortensen</a>.
This notifies to search engines that the text being linked with is the author, and the URL is the author's page.
The author's page must reside on the same website as the author's content. So it doesn't make much sense to link to the author's page from a site where there is no content written by you.
You can link from your author page on one site to your other author pages on other websites by using rel="me" instead of rel="author".
See here for more in-depth info: http://searchengineland.com/google-adds-authorship-rich-snippet-markup-80455
As for how you can use it to build your reputation, you're notifying Google what articles you've written, and where, as well us helping Google find information about you as an author. So next time someone searches for "Peter Mortensen", it may just help you collate all the data together so Google knows this stuff was all written by the same person.
Edit: An interesting addendum to make. While digging around, I found an interview with Matt Cutts and Othar Hansson talking about it, and Hansson suggests that the rel="author" tag may also introduce a signal for credibility, and increasing the ranking of a website: "we hope to use this information and any information as a ranking signal at Google…in this case we’d want to get information on the credibility of authors, from all kinds of sources, and eventually we’ll use that in ranking. We’re only experimenting with that now, who knows where it will go".
Also worth mentioning is that authors' photos appear in Google search results.