为什么仿射平面中的抛物线 y=x² 与直线 y=0 的交点按“计重数”算是 2?这个重数在代数上如何体现?
Why does the parabola y=x² meet the line y=0 in the affine plane with intersection multiplicity 2, and where does that multiplicity appear algebraically?
把直线代入抛物线得到 x²=0,x=0 处出现二重根。代数几何里交点重数定义为局部环对理想商的维数:k[x,y]/(y-x²,y) ≅ k[x]/(x²),作为 k-向量空间是 2 维,故重数为 2。
Substituting the line into the parabola gives x²=0, a double root at x=0. In algebraic geometry the multiplicity is the dimension of the local quotient by the ideal: k[x,y]/(y-x²,y) ≅ k[x]/(x²), which is 2-dimensional over k, so the multiplicity is 2.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
Karen Smith 等《An Invitation to Algebraic Geometry》(Birkhäuser,1968)中关于相交重数与维数的相关章节
Karen E. Smith et al., An Invitation to Algebraic Geometry (Birkhäuser, 1968), chapters on intersection multiplicity and dimension
#交点重数 #局部环
射影平面 P² 中平行直线为何恰好相交于一点?这与仿射平面的差别根源是什么?
In the projective plane P², why do parallel lines meet in exactly one point, and what accounts for the difference from the affine plane?
P² 由 k³ 中的过原点直线构成,点用齐次坐标 [x:y:z] 表示;两条直线 l₁x+m₁y+n₁z=0 与 l₂x+m₂y+n₂z=0 的联立方程总有非零解,故必相交。仿射平面中平行线方向相同但无公共有限点,射影化补上了唯一的无穷远点 [m:−l:0],正是添加这些无穷远点消除了平行带来的例外情形。
P² consists of lines through the origin in k³ with points written [x:y:z]; two lines l₁x+m₁y+n₁z=0 and l₂x+m₂y+n₂z=0 always admit a nonzero common solution, so they intersect. In the affine plane parallel lines share a direction but no finite point; projectivization adds the unique point at infinity [m:−l:0], and adding these points removes exactly the exceptional behavior caused by parallelism.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Cox、Little 与 O'Shea《Ideals, Varieties, and Algorithms》第 8 章关于射影空间的齐次坐标讨论
D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, Chapter 8 on homogeneous coordinates in projective space
独立核验依据 · Independent verification: 核验依据 1 ISBN 9783540978473 (2016)
#射影平面 #齐次坐标
代数几何中的仿射簇是如何定义的？希尔伯特零点定理为什么说“几何”与“理想”可以互相翻译？
How are affine varieties defined in algebraic geometry, and why does Hilbert's Nullstellensatz let us translate between geometry and ideals?
仿射簇是仿射空间中一组多项式公共零点构成的集合。希尔伯特零点定理建立了反变的双向对应：簇对应其消去理想的根理想，而代数闭域上每个根理想都恰是某簇的理想；于是关于零点集的几何问题原则上都可化为交换代数中理想的运算问题。
An affine variety is the set of common zeros of a collection of polynomials in affine space. Hilbert's Nullstellensatz sets up an antitone correspondence: a variety corresponds to the radical ideal of polynomials vanishing on it, and every radical ideal over an algebraically closed field arises this way—so geometric questions about zero sets translate, in principle, into ideal computations in commutative algebra.
难度 进阶 · 适合 大众
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：此条未指向带版次的可引用著作（通识性内容、机构文件或未展开的英文书名），本库不提供独立书目出处；如需精确引用，需由编辑补全具体篇目与版次。
Fulton, W. (1969). Algebraic Curves: An Introduction to Algebraic Geometry. W. A. Benjamin, Chapter 1.
Fulton, W. (1969). Algebraic Curves: An Introduction to Algebraic Geometry. W. A. Benjamin, Chapter 1.
#仿射簇 #零点定理