If you discuss the Whitney umbrella, I guess it is to show that blowing-up the singular point resolves does not resolve the singularity whereas blowing-up the double line doesnot. Another interesting properties of blow-up can be discussed by considering the 3 different resolutions of the conifold $x_1 x_2- x_3 x_4 =0$ in $\mathbb{C}^4$, namely the two small resolutions related by a flop and the blow-up of the isolated singularity with exceptional locus $\mathbb{P}^1\times\mathbb{P}^1$.

A quadric surface $Q$ in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ via the Segre embedding. However, the Zariski topology of $Q$ is not homeomorphic to the product topology of $\mathbb{P}^1\times \mathbb{P}^1$ when each $\mathbb{P}^1$ is considered with the Zariski topology.

The orbifolds $\mathbb{C}^2/ \Gamma$ where $\Gamma$ is a discrete group of $SU(2)$. These orbifolds can be expressed as the simple isolated singularities of a surface and their resolution gives all the ADE Dynkin diagrams.

If you discuss the Whitney umbrella, I guess it is to show that blowing-up the singular point resolves the singularity whereas blowing-up the double line does not. Another interesting properties of blow-up can be discussed by considering the 3 different resolutions of the conifold $x_1 x_2- x_3 x_4 =0$ in $\mathbb{C}^4$, namely the two small resolutions related by a flop and the blow-up of the isolated singularity with exceptional locus $\mathbb{P}^1\times\mathbb{P}^1$.

A quadric surface $Q$ in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ via the Segre embedding. However, the Zariski topology of $Q$ is not homeomorphic to the product topology of $\mathbb{P}^1\times \mathbb{P}^1$ when each $\mathbb{P}^1$ is considered with the Zariski topology.

The orbifolds $\mathbb{C}^3/ \mathbb{C}^2/ \Gamma$ where $\Gamma$ is a discrete group of $SU(2)$. These orbifolds can be expressed as the simple isolated singularities of a surface and their resolution gives all the ADE Dynkin diagrams.

If you discuss the Whitney umbrella, I guess it is to show that blowing-up the singular point resolves the singularity whereas blowing-up the double line does not. Another interesting properties of blow-up can be discussed by considering the 3 different resolutions of the conifold $x_1 x_2- x_3 x_4 =0$ in $\mathbb{C}^4$, namely the two small resolutions related by a flop and the blow-up of the isolated singularity with exceptional locus $\mathbb{P}^1\times\mathbb{P}^1$.

A quadric surface $Q$ in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ via the Segre embedding. However, the Zariski topology of $Q$ is not homeomorphic to the product topology of $\mathbb{P}^1\times \mathbb{P}^1$ when each $\mathbb{P}^1$ is considered with the Zariski topology.

The orbifolds $\mathbb{C}^3/ \Gamma$ where $\Gamma$ is a discrete group of $SU(2)$. These orbifolds can be expressed as the simple isolated singularities of a surface and their resolution gives all the ADE Dynkin diagrams.