# ag.algebraic geometry – How to distinguish the singularities on moduli space?

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.

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