課程名稱︰幾何概論(二)
課程教師︰蔡宜洵
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰99/4/23
考試時限（分鐘）：2 hr
是否需發放獎勵金：是
（如未明確表示，則不予發放）
course: introduction to geometrical thought and methods(2)
midterm examination,April 23 ,2010
total points : 110
1 (25 pts).Let f(x)=x^n + a1x^(n-1)+...an ,ai屬於C (1)(10 pts)Discuss how
the discriminant of f ,denoted by D ,can be defined in terms of coefficients ai's
(2)(15 pts)Show, by using theory of resultant (without giving proof ,but any
theorems needed in your proof should be stated clearly), that D=0 if and only
if f(x) has a multiple zero.
2.(30 pts). Consider P^2 by coordinates (x:y:z) as a union of the affine part
{z≠0} and its line at infinity L :{z=0}.Let C1':y^2=x^3 and C2':3x^2-y^2=2x
be the affine parts of two projective curves C1 and C2 respectively.(1)(10 pts
)First find the equations of C1 and C2 .Then find the points of intersection
C1˙L and C2˙L (2)(10 pts) Find points of intersection C1˙ C2.(3)(10 pts)
Find the local intersection multiplicity for each point of intersection in (1)
and (2) (Note :theorem(s), if any ,used for purpose of evaluation,should be
stated clearly).
3.(30 pts). Let p 屬於C be a singular point of multiplicity r (r>=2), the
same as p be a r-fold point of C .(1)(10 pts) Give the algebraic definition
of the above statement about " r-fold point ".(2)(10 pts) Give a geometrical
interpretation of (1) via intersection with straight lines. (3) (10 pts)
Similar questions for tangents at p ,namely first give the algebraic
definition of tangents at p ,then the geometrical interpretation from the
viewpoint of intersection theory.
4.(25 pts). Let p 屬於C.(1)(10 pts) Suppose p is a simple point of C .By
suitable choice of coordinates ,state (without giving proof) the "implicit
function theorem for formal power series .(2)(15 pts)Suppose p is an ordinary
singular point of multiplicity r (r>=2). Show ,by using (1) and suitable
coordinates ,that there exists exactly r different series expansion around p.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.216.23