When computing the dimension of moduli space for complete intersections of type $(a,b)$ in $\mathbb{P}^n$, what do we need to consider? In general we have the following part:
$$|\mathcal{O}\_{\mathbb{P}^n}(a)|+|\mathcal{O}\_{\mathbb{P}^n}(b)|-\text{PGL}(n+1)$$
then what should one eliminate again? Here some concrete examples.



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> For hypersurfaces, we do not need to minus something, the moduli space is *roughly* $|\mathcal{O}\_{\mathbb{P}^n}(b)|/\text{PGL}(n+1)$
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> For complete intersections of type $(2,3)$ in $\mathbb{P}^4$, we need to minus an extra family of cubics. Why we need this?
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> In this computation [$(2,4)$ type in $\mathbb{P}^5$](https://math.stackexchange.com/questions/807687/dimension-of-some-moduli-spaces/817836#817836), they minus a whole extra family of quadrics.
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The (expected) dimension of moduli space for complete intersection
First you want to compute $h^0(\mathcal{O}\_X(a) \oplus\mathcal{O}\_X(b))$, if $X$ is your complete intersection. You can do it by writing down the Koszul complex for $X$ and tensoring it with $\mathcal{O}\_X(a) \oplus\mathcal{O}\_X(b)$.
Using the properties of the Euler characteristic, you can find the desired dimension,
$$
h^0(\mathcal{O}\_X(a) \oplus\mathcal{O}\_X(b)) = \chi(\mathcal{O}\_\mathbb{P}(a))+\chi(\mathcal{O}\_\mathbb{P}(b)) + \chi(\mathcal{O}\_\mathbb{P}(-a))+\chi(\mathcal{O}\_\mathbb{P}(-b)) -2 - \chi(\mathcal{O}\_\mathbb{P}(b-a)) - \chi(\mathcal{O}\_\mathbb{P}(a-b))
$$


You can easily compute the above values. For example, for a $2,3$ complete intersection in $\mathbb{P}^4$, you have
$$
h^0(\mathcal{O}\_X(2) \oplus\mathcal{O}\_X(3))= {2+4 \choose 4}+{3+4 \choose 4}- 2- {1+4 \choose 4}=43
$$
To get your desired number, you need to take $(n+1)^2-1$ out (the dimension of the projective linear group). In this case it is 24, hence $43-24=19$.


