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In Harris' and Morrison's book "Moduli of Curves" page 123, they study stable reduction of cuspidal singularity appearing in the center fiber. When they first blow up the total family at the origin, why does the exceptional divisor $E_1$ have multiplicity two? Thank you for your help.

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If we look at the example of the pencil of curves $y^2-x^3-t = 0$ over $\mathbb C$ with parameter $t$, we can see where the multiplicity comes from explicitly. Consider the family as the hypersurface $\operatorname{Spec}(\mathbb C[x,y,t]/(y^2-x^3-t))$ of $\mathbb A^3$. We can compute a chart of the blowup, say the one determined by $$\left(\dfrac{\mathbb C[x,y,t]}{(y^2-x^3-t)}\right)[y/x,t/x] = \mathbb C[x,y/x,t/x]/\left((x\cdot y/x)^2 - x^3 - (x\cdot t/x)\right).$$ The local equation for the blown up pencil in this chart is $(x\cdot y/x)^2 - x^3 - (x\cdot t/x)$, and we see that computing its fibre over $t = 0$ gives $(x\cdot y/x)^2 - x^3 = x^2((y/x)^2 - x)$, which has two components: the exceptional divisor $E\colon(x^2 = 0)$, and the normalization of the cusp $\widetilde C\colon ((y/x)^2 - x = 0)$.

Similarly, we can compute $E:(y^2 = 0)$ and $\widetilde C\colon (1-(x/y)^3y)$ in the $y$-chart. Either way, the exceptional divisor comes with multiplicity two.

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  • $\begingroup$ Dear @MichaelJoyce, thank you, I certainly appreciate the kind words! $\endgroup$
    – Andrew
    Aug 28, 2013 at 19:55
  • $\begingroup$ Thank you Andrew, it is really very clear. $\endgroup$
    – JacobI
    Aug 29, 2013 at 6:31

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