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Let $f\colon X\to \mathbb{P}^3$ be a finite morphism, where $X$ is a smooth and irreducible algebraic variety of dimension three (everything over $\mathbb{C}$).

The ramification locus of $f$ is the closed subset where the differential of $f$ is not an isomorphism, or, in other words:

$$ R_2f = \{ x\in X \,|\, \operatorname{rank} df_x \leq 2 \} $$

Since the map is finite, we know that the ramification locus has codimension at least one, i.e. it is either empty, or a divisor.

What I am interested in are higher ramification loci, and especially:

$$ R_1f = \{ x\in X \,|\, \operatorname{rank} df_x \leq 1 \}$$

Question: Are there any conditions on $f$ which give a lower bound on the codimension of $R_1f$?

For example, we can look at the differential of $X$ as a morphism of vector bundles $df\colon TX \to f^*T{\mathbb{P}^3}$ of rank three on $X$, and then $R_2f$ is the locus where this morphism has rank at most one. In my particular case, I know that this map is symmetric, hence the condition of being of rank at most one is represented by the vanishing of all the $2\times 2$ principal symmetric minors of $df$. Thus, we expect $R_1f$ to have codimension three. Are there any conditions that guarantee that the codimension is exactly three?

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The cohomology class of the expected corank 2 locus for a morphism of vector bundles is given by the Porteous formula (there is also a version of that formula for symmetric maps). If this class is nonzero, then the corank 2 locus is nonempty.

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  • $\begingroup$ Thank you Sasha. Indeed, I was aware of this, and in my case, I get a nonzero cohomology class. However my problem is more that the locus could be too big, hence I am looking at a lower bound for the codimension. Is there any way to get one? $\endgroup$
    – Daniele A
    Commented Apr 23, 2019 at 13:50
  • $\begingroup$ If $\phi \colon E \to F$ is a general morphism of vector bundles (i.e., general element in the vector space $\Hom(E,F)$) and if the bundle $E^\vee \otimes F$ is globally generated, then the corank 2 locus has expected codimension. Does that help? $\endgroup$
    – Sasha
    Commented Apr 23, 2019 at 17:14
  • $\begingroup$ Thank you for the suggestion. Unfortunately, I think that this does not help since the morphism is not general: is the one induced by the differential of $f$. $\endgroup$
    – Daniele A
    Commented Apr 25, 2019 at 13:26

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