3 edited body

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a cusp node at $(0,0)$ (and at some other points) (it's not a homework question, just a concept that I'm stuck on!). $xy=x^6+y^6$. Then the blow-up should be the closure of this set, taken over all $(x,y) \neq (0,0)$: ${ ((u,v), (x,y)) \in \mathbb{A}^{2} \times \mathbb{P}^{1} | uy=vx, xy = x^6 + y^6 }$.

How do I explicitly find the closure of this set? I understand the fibre of the projection map at the singular point should consist of two points (both of which are non-singular) - why is this so, and what are those two non-singular points (in the smooth variety that is the resolution)?.

2 corrected projective plane to projective line

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a cusp at $(0,0)$ (and at some other points) (it's not a homework question, just a concept that I'm stuck on!). $xy=x^6+y^6$. Then the blow-up should be the closure of this set, taken over all $(x,y) \neq (0,0)$: ${ ((u,v), (x,y)) \in \mathbb{A}^{2} \times \mathbb{P}^{2} mathbb{P}^{1} | uy=vx, xy = x^6 + y^6 }$.

How do I explicitly find the closure of this set? I understand the fibre of the projection map at the singular point should consist of two points (both of which are non-singular) - why is this so, and what are those two non-singular points (in the smooth variety that is the resolution)?.

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finding the closure when blowing a variety at a singularity

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a cusp at $(0,0)$ (and at some other points) (it's not a homework question, just a concept that I'm stuck on!). $xy=x^6+y^6$. Then the blow-up should be the closure of this set, taken over all $(x,y) \neq (0,0)$: ${ ((u,v), (x,y)) \in \mathbb{A}^{2} \times \mathbb{P}^{2} | uy=vx, xy = x^6 + y^6 }$.

How do I explicitly find the closure of this set? I understand the fibre of the projection map at the singular point should consist of two points (both of which are non-singular) - why is this so, and what are those two non-singular points (in the smooth variety that is the resolution)?.