Configuration of infinitely near points

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Despite the fact that Hartshorne (in Algebraic Geometry, Ch. 5, Exercise 3.8) asserts that the singularities of the plane curves



$$x^4-xy^4=0$$
and
$$x^4-x^2y^3-x^2y^5+y^8=0,$$
have "the same configuration of infinitely near singular points with the same multiplicities", it seems to me that after a single blowup the new first curve possesses a single singular point of multiplicity $2$, while the new second curve possesses a single singular point of multiplicity $3$; who (if anyone) is correct?




algebraic-geometry




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  • It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
    – Asal Beag Dubh
    Jul 31 at 8:46










  • For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
    – dOuUuq3podCuoArv
    Jul 31 at 14:38










  • You are right; I made an error in my calculation.
    – Asal Beag Dubh
    Jul 31 at 14:50














up vote
1
down vote

favorite












Despite the fact that Hartshorne (in Algebraic Geometry, Ch. 5, Exercise 3.8) asserts that the singularities of the plane curves



$$x^4-xy^4=0$$
and
$$x^4-x^2y^3-x^2y^5+y^8=0,$$
have "the same configuration of infinitely near singular points with the same multiplicities", it seems to me that after a single blowup the new first curve possesses a single singular point of multiplicity $2$, while the new second curve possesses a single singular point of multiplicity $3$; who (if anyone) is correct?




algebraic-geometry




share|cite|improve this question



















  • It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
    – Asal Beag Dubh
    Jul 31 at 8:46










  • For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
    – dOuUuq3podCuoArv
    Jul 31 at 14:38










  • You are right; I made an error in my calculation.
    – Asal Beag Dubh
    Jul 31 at 14:50












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Despite the fact that Hartshorne (in Algebraic Geometry, Ch. 5, Exercise 3.8) asserts that the singularities of the plane curves



$$x^4-xy^4=0$$
and
$$x^4-x^2y^3-x^2y^5+y^8=0,$$
have "the same configuration of infinitely near singular points with the same multiplicities", it seems to me that after a single blowup the new first curve possesses a single singular point of multiplicity $2$, while the new second curve possesses a single singular point of multiplicity $3$; who (if anyone) is correct?




algebraic-geometry




share|cite|improve this question











Despite the fact that Hartshorne (in Algebraic Geometry, Ch. 5, Exercise 3.8) asserts that the singularities of the plane curves



$$x^4-xy^4=0$$
and
$$x^4-x^2y^3-x^2y^5+y^8=0,$$
have "the same configuration of infinitely near singular points with the same multiplicities", it seems to me that after a single blowup the new first curve possesses a single singular point of multiplicity $2$, while the new second curve possesses a single singular point of multiplicity $3$; who (if anyone) is correct?





algebraic-geometry





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 23:13









dOuUuq3podCuoArv

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  • It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
    – Asal Beag Dubh
    Jul 31 at 8:46










  • For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
    – dOuUuq3podCuoArv
    Jul 31 at 14:38










  • You are right; I made an error in my calculation.
    – Asal Beag Dubh
    Jul 31 at 14:50
















  • It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
    – Asal Beag Dubh
    Jul 31 at 8:46










  • For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
    – dOuUuq3podCuoArv
    Jul 31 at 14:38










  • You are right; I made an error in my calculation.
    – Asal Beag Dubh
    Jul 31 at 14:50















It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
– Asal Beag Dubh
Jul 31 at 8:46




It appears to me that after one blowup, the proper transforms of both curves have a singular point of multiplicity 3.
– Asal Beag Dubh
Jul 31 at 8:46












For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
– dOuUuq3podCuoArv
Jul 31 at 14:38




For example, to get the blowup of the first curve (in one of the charts), I made the substitution $(x,y) mapsto (xy,y)$, yielding the equation $x^4y^4-xy^5$; the proper transform has equation $x^4-xy$. The singular point $(0,0)$ has multiplicity $2$.
– dOuUuq3podCuoArv
Jul 31 at 14:38












You are right; I made an error in my calculation.
– Asal Beag Dubh
Jul 31 at 14:50




You are right; I made an error in my calculation.
– Asal Beag Dubh
Jul 31 at 14:50















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