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Consider a surface $X$ (which is a projective variety over $\mathbb{C}$) with a double point singularity. To make $X$ smooth, we can either blow up the singularity with a $(-2)$ - curve ($\hat{X}$), or we can simply deform the variety's defining relation ($\tilde{X}$).

In principle, $\hat{X}$ and $\tilde{X}$ are birational to each other (we can find a Zariski open set in both of them that are isomorphic), so we can start from either of them and after a sequence of blow-ups and blow downs get the other one.

I'm wondering whether it's possible to find this sequence of blow-ups and blow downs explicitly...

Maybe it's difficult to answer generally, but I appreciate if you let me know any example that you are aware of...

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    $\begingroup$ If you deform a nodal quartic in $\mathbb{P}^3$ to a smooth quartic, the resulting surface will be in general not birational to the nodal one. In fact, you can deform the nodal quartic to every smooth quartic, since the deformation space of quartics is connected. And the moduli space of quartics has dimension $19$. $\endgroup$
    – Francesco Polizzi
    Jun 15, 2018 at 21:58
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    $\begingroup$ In general, it is false that deforming equations produce birational varieties. Typical example: a nodal cubic curve is rational, but you can deform it to a smooth cubic curve, that has genus $1$. $\endgroup$
    – Francesco Polizzi
    Jun 15, 2018 at 22:01
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    $\begingroup$ (1) This is a general fact about moduli of surfaces in $\mathbb{P}^3$, you can have a look in Sernesi book on deformation theory. (2) The example of cubic curves in fact shows that the geometric genus is not deformation invariant (but it is a birational invariant). Concerning the arithmetic genus, it is both deformation and birationally invariant. $\endgroup$
    – Francesco Polizzi
    Jun 15, 2018 at 23:08
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    $\begingroup$ Since you are a student in string theory, maybe it would be worth framing this as follows: smoothing the node is changing complex moduli, resolving the node is changing Kahler moduli. These two operations are in some sense dual to each other under mirror symmetry. This is literally true one dimension higher where one can either resolve or smooth a conifold singularity --- you are describing the 2D version of a conifold transition. $\endgroup$
    – Jim Bryan
    Jun 18, 2018 at 17:59
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    $\begingroup$ One statement about derived categories which is true is the following. Let $X$ be the surface with the double point singularity, let $\hat{X}$ be is its minimal resolution, and let $\mathcal{X}$ be the orbifold (i.e. smooth Deligne-Mumford stack) whose coarse space is $X$. Then $D(\hat{X})$ is equivalent to $D(\mathcal{X})$. $\endgroup$
    – Jim Bryan
    Jun 19, 2018 at 16:30

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