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Let $X$ be a smooth Gushel-Mukai threefold, there are following four cases:

  1. $X_1$ is a special Gushel-Mukai with branch locus $\mathcal{B}$ on $Y_5$ general, i.e, it does contain any line or conic. $\pi:X\rightarrow Y_5$ is a covering map.
  2. $X_2$ is a special Gushel-Mukai with branch locus $\mathcal{B}$ contain lines or conics.
  3. $X_3$ is a general ordinary Gushel-Mukai.
  4. $X_4$ is a non-general ordinary Gushel-Mukai.

Now, I consider their Fano surface of conics, denoted by $F_c(X_i)$.

  1. $F_c(X_1)$ is a reducible surface with two irreducible components and their intersection is a curve parameterized by $\pi$-preimage of a smooth rational curves $\rho$ parametrizing the $(-1,1)$-lines on $Y_5$ and its singular locus is $\pi$-preimage of $\rho$.
  2. $F_c(X_2)$ has extra singularities given by $\pi$-preimage of finitely many lines in $\mathcal{B}$ and finitley many conics in $\mathcal{B}$ .
  3. $F_c(X_3)$ is a smooth irreducible surface.

My question What about $F_c(X_4)$?, it is still irreducible, but can it be smooth? The normal bundle of a conic on $X_4$ can only be $(0,0), (-1,1)$ and $(-2,2)$-type. If it is singular, is it possible that the singular locus of $F_c(X_4)$ contains a curve?

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