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Let f(x,y) = x^2 y - xy = x(x-1)y be a polynomial in k[x,y].

I am looking for the singular subset of this function.

Taking the partials, we obtain

f_x = 2xy - y

f_y = x^2 - x.

In order to find the singular subset, both partials (with respect to x and with respect to y) must vanish. So we obtain that

f_x = 2xy - y = 0

which implies x = 1/2 or y = 0,

while

f_y = x^2 - x = 0

implies

x = 0 or x =1.

Drawing a picture of f, it is clear that the two points (0,0) and (1,0) are singular points, but what does x = 1/2 tell us? Is this point supposed to be a singular point as well?

I am looking for the singular subset of this function.

Taking the partials, we obtain

f_x = 2xy - y

f_y = x^2 - x.

In order to find the singular subset, both partials (with respect to x and with respect to y) must vanish. So we obtain that

f_x = 2xy - y = 0

which implies x = 1/2 or y = 0,

while

f_y = x^2 - x = 0

implies

x = 0 or x =1.

Drawing a picture of f, it is clear that the two points (0,0) and (1,0) are singular points, but what does x = 1/2 tell us? Is this point supposed to be a singular point as well?

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