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Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.

Let $f:X\longrightarrow Y$ be a minimal resolution of singularities. Assume that $X$ is semi-stable over $S$. In particular, $X$ is minimal and doesn't contain any $(-1)$-curves.

Let $y$ be a singular point of $Y$ and let $E_1,\ldots,E_r$ be the exceptional components of $f$ lying over $y$.

Fix $j \in \{1,\ldots,r\}$.

Do we have that $$ \sum_{i=1}^r (E_i,E_j) \leq 0?$$

Here $(\cdot, \cdot)$ denotes the intersection pairing on the regular fibered surface $X$.

For any $j\in \{1,\ldots,r\}$, I know that $$ X_y \cdot E_j = \sum_{i=1}^r d_i (E_i,E_j) \leq 0,$$ where $d_i$ is the multiplicity of $E_i$.

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    $\begingroup$ @Frederick : You assume that $X$ is semi-stable - doesn't that imply (by definition) that $d_i=1$ for all $i$ ? $\endgroup$
    – Damian Rössler
    Aug 27, 2011 at 12:32
  • $\begingroup$ Yes! You're completely right. I can't believe I missed that...thank you. $\endgroup$
    – Frederick
    Aug 27, 2011 at 20:46

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