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 $mathcal{C}(X)=overline{mathcal{F}}cuppi^*Sigma(Y)$, 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 $pi:Xrightarrow Y_5$ 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 $pi^*(rho)cuppi^*{Lsubsetmathcal{B}}$, 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 $pi^*Sigma(Y)congmathbb{P}^2$ 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 $mathcal{N}_{C|X’}=mathcal{O}_C(2)oplusmathcal{O}_C(-2)$(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 $Esubsetmathcal{C}(X’)$ 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.