I believe that the following questions are very basic, but I don't know how to get a reference. 


Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is 
unibranch at zero (i.e. analytically irreducible). Then I guess one should be able to define "arithmetic genus defect" of the curve at $0$. Namely if one smooths analytically $C$, its geometric genus will grow by a positive number (in case of the cusp $x^2=y^3$ it will grow by one), and let us call this number the defect. 


**Question 1.** Is this defect well defined (independent of a smoothing)? How is it called and how one should calculate it (say it terms of the local ring of $C$ at $0$)?


**Question 2.** Suppose we have an explicit local parametrisation of $C$ at $0$, say by two holomorphic functions $f(t), g(t)$ (polynomials if you wish). Is it possible to find this "defect" as a certain invariant of this pair of functions at $t=0$? 


*Question 1 is settled in the answer of unknown and Question 2 in comments to it by Roy and Vivek*



"Arithmetic genus" of a plane curve singularity.
The difference between the geometric genus of the singularity and the geometric genus of a
smoothing (this one being called the arithmetic genus of the singularity) is often called
the *delta invariant*. If $A$ is the local ring of the singularity, $B$ its normalization, then the delta invariant is the dimension of the complex vector space $B/A$.


It is rather easy to compute the delta invariant if one knows an equation $f(x,y)=0$ of the curve by a formula due to Milnor (see the book "Singularities of hypersurfaces"):
 $2 \delta = \mu + b - 1 $ where $\delta$ is the delta invariant, $\mu = \dim\_{\mathbb{C}} \mathbb{C}[[x,y]]/(\partial\_{x}f, \partial\_{y}f )$ and $b$ is the number of branches. In the unibranch case, it is simply $2 \delta = \mu$ (example: for the cusp, $\delta = 1$, $\mu =2$).


