Is sum (E_i, E_j) non-positive, with E_i's the exceptional components of a desingularization - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:13:44Z http://mathoverflow.net/feeds/question/73838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73838/is-sum-e-i-e-j-non-positive-with-e-is-the-exceptional-components-of-a-desing Is sum (E_i, E_j) non-positive, with E_i's the exceptional components of a desingularization Frederick 2011-08-27T11:51:34Z 2011-08-27T11:51:34Z <p>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.</p> <p>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.</p> <p>Let $y$ be a singular point of $Y$ and let $E_1,\ldots,E_r$ be the exceptional components of $f$ lying over $y$.</p> <p>Fix <code>$j \in \{1,\ldots,r\}$</code>. 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 $X$.</p> <p>For any <code>$j\in \{1,\ldots,r\}$</code>, 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 $E_i$. </p>