What's the purpose of
3. Nd2 in the French opening? Does it not block the dark-squared bishop and offer space in center for black? What's the sharpest line for white in this variation?
[FEN ""] 1. e4 e6 2. d4 d5 3. Nd2
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Now I haven't played the Tarrasch, but I am considering it, as an alternative to the exchange variation, and this is what I've discovered in my limited research.
The main purpose is to support the pawn chain, if Black continues with
3. ... Nf6, as the c3 square is available for a pawn to support d4. Additionally, as a bonus, it avoids the Winawer variation of the French, without playing the advanced or exchange variations, which are easier for black at higher levels.
While it does block in the Queen's Bishop, other than the classical lines,
1. e4 e6 2.d4 d5 3. Nc3 Nf6 4. Bg5, it's not uncommon for the Queen Bishop to remain undeveloped for most of the opening.
As for which line is the sharpest, a lot depends on black's responses.
The main purpose of 3. Nd2, as opposed to 3. Nc3, is to avoid the pin after 3...Bb4. For inspiration you might check out Kasparov's games. The Tarrasch was his main weapon against the French for years and he scored many crushing victories. For example...