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Let $\mathcal E$ be a rank $2$ vector bundle on a smooth (complex) variety $X$ and $\mathcal L$ a line bundle.
Consider $q\in H^0(X,{\rm Sym}^d\mathcal E\otimes \mathcal L)$ ($d>2$) a polynomial form on $\mathcal E^*$.
How can we compute the line bundle the discriminant of $q$ is a section of (i.e. the class of the locus $\{x\in X,\ \{q_x=0\}\subset \mathbb P(\mathcal E_x)\ {\rm is\ not\ smooth}\}$)?

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    $\begingroup$ What is $q'_x$? $\endgroup$
    – abx
    Nov 8, 2022 at 19:13
  • $\begingroup$ Perhaps I should not write it thus. It stands for the derivative of $q$ locally (so that it is a polynomial in one variable) in $\mathbb P(\mathcal E)$ $\endgroup$
    – pi_1
    Nov 8, 2022 at 19:57

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