Let me start with concrete examples. Let $X$ be a smooth special Gushel-Mukai threefold and $\mathcal{C}(X)$ be its honest Fano surface of conics, it has two irreducible components $\mathcal{C}(X)=\overline{\mathcal{F}}\cup\pi^*\Sigma(Y)$, where $\overline{\mathcal{F}}$ is the closure of double cover of family conics which are double tangent to the branch locus $\mathcal{B}$ on $Y_5$ and $\Sigma(Y)$ is the Fano surface of lines on $Y_5$, the map $\pi:X\rightarrow Y_5$ is a double cover with branch locus $\mathcal{B}$. Note that the component $\pi^*\Sigma(Y)$ is a $\mathbb{P}^2$. Also we have the intersection of two components is $\pi^*(\rho)\cup\pi^*\{L\subset\mathcal{B}\}$, where $\rho$ is the curve of $(-1,1)$-lines on $Y$ and $L$ is line in branch locus $\mathcal{B}$.
Let $\mathcal{C}_m(X)$ be the contraction of $\mathcal{C}(X)$ along the component $\pi^*\Sigma(Y)\cong\mathbb{P}^2$ to a point. Then $\mathcal{C}_m(X)$ becomes irreducible.
First, I assume that $X$ is general in the sense that $\mathcal{B}$ is general, i.e. does not contain any line or conic. Then one can show that $\mathcal{C}_m(X)$ has the unique singular point, denoted by $q$, I call this $q$ a type $I$ singularity. Now I assume $X$ is not general but with branch locus only contains one line $L$, then $\mathcal{C}_m(X)$ still has a unique singularity $q'$ because $\pi^*L$ is still in the component $\pi^*\Sigma(Y)$ and they contract to one point, but I think in this case the singularity type of $q'$ should be different with the previous case, I call it type $II$. Now, I assume that the branch locus $\mathcal{B}$ could also contain conic. Then $\mathcal{C}_m(X)$ would have extra isolated singular point $q''$ given by those conics, since they are fixed by the geometric involution $\tau$(coming from the double cover), I call those singularity Type $III$.
My first question is are these three type singularity are of the same type or not? For example, how to tell what kind of singularity type do they have, say $A_1, A_2$ etc? I would guess that type I and type II are different but for type I and type III, I am not so sure.
Consider the similar story on ordinary Gushel-Mukai threefold $X'$, the honest Fano surface of conics $\mathcal{C}(X')$, this is always irreducible surface. If $X'$ is general, then $\mathcal{C}(X')$ is smooth and if $X'$ is non-general, then there are singular points given by $\tau$-conic $C$ with normal bundle $\mathcal{N}_{C|X'}=\mathcal{O}_C(2)\oplus\mathcal{O}_C(-2)$(or equivalently, such conic is fixed by some involution $\tau$, but this involution appears in the blow up of $\mathcal{C}(X')$ or blow down of $\mathcal{C}(X')$). It is known that there is a unique exceptional curve $E\subset\mathcal{C}(X')$ consists of $\sigma$-conics and if we contract this exceptional curve we get the minimal surface of general type $\mathcal{C}_m(X')$. If $X'$ is general, then it is smooth, if $X'$ is not general, then it has singularity $q'''$ given by the $\tau$-conic I mentioned above, in this case, I call singularity type of $q'''$ type $IV$.
My second question: Is this type $IV$ singularity type the same as type $I, III, III$? My third question: how to study these singularities in the moduli space? I tried to google, but end up with nothing useful.
