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Let $X$ be a smooth (complex) projective $n$-dimensional variety ($n\geq 3$) and $\mathcal E$ a vector bundle of rank $r<n$ generated by its global sections on $X$. Let $\sigma\in H^0(\mathcal E)$ be a section and $\mu:X\rightarrow P(\mathcal E)$ a section of the projective bundle. Is there a way to express, in term of chern classes of $\mathcal E$ and $\mu^*\mathcal O_{P(\mathcal E)}(1)$, the class in $X$ of the locus (which contains the zero locus of $\sigma$) where $span(\sigma,\mu)$ is one dimensional ?

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  • $\begingroup$ Aside from the fact that you're making an assumption that $\mu$ exists, you don't really care that it's nonvanishing, do you? In which case this is about a map from the trivial 2-plane bundle into $\mathcal E$, asking where its rank is only $1$. That's the domain of the Thom-Porteous formula. en.wikipedia.org/wiki/Porteous_formula $\endgroup$ – Allen Knutson Aug 13 '16 at 17:38
  • $\begingroup$ Thank you. The problem with $\mu$ is that it is determined only up to a scalar so that (I could be mistaken) the morphism of from the trivial bundle to $\mathcal E$ is not well defined, isn't it? $\endgroup$ – pi_1 Aug 13 '16 at 17:47
  • $\begingroup$ Oh, I see. Trying (and failing) to fix that scalar gives you the transition functions for the line bundle $\mathcal L := \mu^*\mathcal O_{P(\mathcal E)}(1)$, I guess. So then it's about a map from $\mathcal O \oplus \mathcal L$ into $\mathcal E$, and still determined by Thom-Porteous. $\endgroup$ – Allen Knutson Aug 13 '16 at 18:36

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