Problems for next time.

1. (Jeff) Let $\P ^2$ be the set of triples $[a,b,c]$ modulo scalar multiplication, as usual. Define a line in $\P ^2$ to be the set of solutions of an equation of the form
   $$a X + b Y + c Z=0$$
   for some numbers $a, b, c$ not all zero. Prove (from the definition) that any two distinct points in $\P ^2$ are contained in a unique line. Then prove that any two distinct lines in $\P ^2$ intersect in a unique point.

2. (Jennifer) Let $F(X,Y,Z) \in {\mathbb{C}}[X,Y,Z]$ be a homogeneous polynomial of degree $n$. Prove that the partial derivatives of $F$ are homogeneous polynomials of degree $n-1$, and use this to show that
   $$X \frac{\partial F}{\partial X} + Y \frac{\partial F}{\partial Y} + Z \frac{\partial F}{\partial Z} = n F$$
   by differentiating $F(tX,tY,tZ) = t^n$ with respect to $t$.

3. (Mauro)
   1. Let $C$ be a curve in $\P ^2$ defined by $F(X,Y,Z)=0$, where $F$ is a homogenous polynomial. Prove that if  $P\in \P ^2$ satifies the equation

      $$\frac{\partial F}{\partial X}(P) = \frac{\partial F}{\partial Y}(P) = \frac{\partial F}{\partial Z}(P) = 0,$$ (1)

      
      
      then $P$ ``automatically'' satisfies $F(P)=0$. Thus to find the singular points on $C$, you just have to find the common solutions to (1); it is not necessary to include $F=0$.

   2. Find all singular points on the curve defined by
      $$F(X,Y,Z) = X^7-Y^2Z^5=0.$$

4. (Alex) For each of the given affine curves $C_0$, find a projective curve $C$ whose affine part is $C_0$. Then find all of the points at infinity on the projective curve $C$.
   1. $3x-7y+5=0$
   2. $x^2+xy-2y^2+x-5y+7=0$
   3. $x^3+x^2y - 3xy^2 - 3y^3+2x^2-x+5=0$

5. (Jenna) For each of the following curves $C$ and points $P$, either find the tangent line to $C$ at $P$ or else verify that $C$ is singular at $P$.

   $C$	$P$
   $y^2=x^3-x$	$(1,0)$
   $X^2+Y^2=Z^2$	$(3:4:5)$
   $x^2+y^4+2xy+2x+2y+1=0$	$(-1,0)$
   $X^3+Y^3+Z^3 = XYZ$	$(1:-1:0)$

6. (Alex) Let $C$ be the cubic curve $u^3+v^3=u+v+1$. In the projective plane, the point $(1:-1:0)$ at infinity lies on this curve. Find rational functoins $x=x(u,v)$ and $y=y(u,v)$ so that $x$ and $y$ satisfy a cubic equation in Weierstrass normal form (i.e., $y^2=x^3+ax^2+bx+c$).

7. (Jeff) Let $C$ be the cubic curve in $\P ^2$ given by
   $$Y^2 Z = X^3 + aX^2Z + bXZ^2 + cZ^3.$$
   Prove that the point $(0:1:0)$ on $C$ is nonsingular.

8. (Jenna) Let $C_1$ and $C_2$ be the cubics given by the following equations:
   $$C_1: x^3+2y^3 - x - 2y = 0,\qquad C_2: 2x^3+y^3 - 2x - y = 0.$$
   Find the nine points of intersection of $C_1$ and $C_2$.

9. (Jennifer) The cubic curve $u^3+v^3=\alpha$ (with $\alpha\neq 0$) has a rational point $(1,-1,0)$ at infinity. Taking this rational point to be $\O$ (the identity element of the group), we can make the points on the curve into a group. Derive a formula for the sum $P_1+P_2$ of two distinct points $P_1=(u_1,v_1)$ and $P_2=(u_2,v_2)$.

10. (Mauro) Verify that if $u$ and $v$ satisfy the relation $u^3+v^3=1$, then the quantities
    $$x = \frac{12}{u+v}$$   and$$\qquad y = 36 \frac{u-v}{u+v}$$
    satisfy the relation $y^2=x^3-432$. We thus obtain a birational transformation $f$ from the curve $u^3+v^3=1$ to the curve $y^2=x^3-432$. Each of these cubic curves can have a group law defined on it. Prove that $f$ is an isomorphism of groups, where the zero element for $y^2=x^3-432$ is the point $(0:1:0)$ and the zero element for $u^3+v^3=1$ is $(1:-1:0)$ (at infinity).