I'm reading the Atanas Atanasov's course notes of Joe Harris' course [Geometry of Algebraic Curves][1] 
and have a question about a suggested modification of an dimension 
counting argument applying methods from deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ 
X \text{ curve of genus } g, f \text{ has degree } d 
\text{ and is birational } \\ \text{ onto a plane curve with }
\delta \text{ nodes } \}   $$

together with two canonical canonical projection maps
$V_{d,g} \to M_g $ (to the 'naive' moduli set) and 
$V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}= 3d+g-1$ 
if $d(d+3)/2 \ge 3 \delta$ but the **Remark 4.2** says:

>There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $
but this can be fixed using deformation theory.

Could somebody elaborate this deformation theoretic argument
fixing the gap the remark 4.2 is refering to?


  [1]: https://staff.math.su.se/shapiro/UIUC/curvesHarris.pdf