In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): suppose $X$ is a complete intersection CY 3-fold (i.e. dualizing sheaf is trivial) in $\mathbb P^{n+3}$ with only ODP's (ordinary double points) as singularities such that the $n$ equations $f_1,...,f_n$ which define $X$ all have the same degree $d$. Assume further that there exists a projective small resolution $\tilde{X}\rightarrow X$ of the singularities of $X$. Notice that $\tilde{X}$ is then a smooth CY 3-fold. Let $S=\mathbb C[x_0,...,x_{n+3}]/(f_1,...,f_n)$ be the homogeneous coordinate ring of $X$.

Now they claim that there is a natural isomorphism $T^1\cong \left(\frac{S^n}{\{(\partial f_1/\partial x_i,...,\partial f_n/\partial x_i)\}}\right)_d$, where the subscript $d$ refers to the degree $d$ homogeneous part of this module and $T^1\cong \operatorname{Ext}^1(\Omega_X^1,\mathcal O_X)$ is the space of infinitesimal deformations of $X$, or equivalent the tangent space to the deformation space of $X$. Why is there such a natural isomorphism?

They further state that the space of infinitesimal deformations of the small resolution $\tilde{X}$ is obtained as the kernel of the natural map $T^1 \rightarrow T^1_{\text{loc}}$, where $T^1_{\text{loc}}$ is the tangent space of the deformation space of the germ of the singular locus of $X$. Why is this the case?

Finally, they remark that this kernel is given by elements $(g_1,...,g_n)\in S^n$ representing an element of $T^1$ such that adding this as a new column to the Jacobian matrix gives a matrix which has rank $<n$ at the singular points of $X$. Why does the kernel have this description?

Thanks