How to compute the Picard-Lefschetz monodromy matrix of a non-semistable fiber? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:24:26Z http://mathoverflow.net/feeds/question/20132 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20132/how-to-compute-the-picard-lefschetz-monodromy-matrix-of-a-non-semistable-fiber How to compute the Picard-Lefschetz monodromy matrix of a non-semistable fiber? Jun Lu 2010-04-02T03:21:21Z 2010-04-02T07:40:39Z <p>Let $f:X\to B$ be a family of curves of genus $g$ over a smooth curve $B$. Let $F_0$ be a singular fiber. </p> <p>If $F_0$ is a semistable fiber, the monodromy matrix can be gotten by the classical Picard-Lefschetz formula. </p> <p>If $F_0$ is non-semistable, I don't know how to compute its monodromy matrix. For example, in Namikawa and Ueno's paper[1], they can compute the Picard-Lefschetz monodromy matrix for each type of singular fiber of genus 2. It's not clear to me how they did that. </p> <p>[1] Namikawa, Y. and Ueno, K., The complete classification of fibres in pencils of curves of genus two, Manuscripta math., Vol. 9 (1973), 143-186.</p> http://mathoverflow.net/questions/20132/how-to-compute-the-picard-lefschetz-monodromy-matrix-of-a-non-semistable-fiber/20137#20137 Answer by Emerton for How to compute the Picard-Lefschetz monodromy matrix of a non-semistable fiber? Emerton 2010-04-02T03:48:59Z 2010-04-02T03:48:59Z <p>One approach (I don't know how effective it is in the genus 2 case you asked about) is to explicitly apply the semi-stable reduction theorem, and so reduce to the semi-stable case. </p> <p>To achieve semi-stable reduction, you have to alternately blow-up singular points in the special fibre, and then make ramified base-changes. The latter operation just extracts a root of the monodromy operator (i.e. if $\gamma$ is a generator of $\pi_1$ of the punctured $t$-disk, and we set $t = s^n$, then $\gamma = \tau^n,$ where $\tau$ is a generator of $\pi_1$ of the punctured $s$-disk), so it is easy to see how the monodromy matrix changes. And blowing up a point in the special fibre doesn't change the monodromy action around the puncture at all.</p> <p>So using this process, one can relate the original (unknown) monodromy matrix to the corresponding matrix in the semi-stable context, where it is known thanks to the Picard--Lefshcetz formula.</p>