1.1 A brief introduction to computational algebraic geometry  (Page 4/20)

If we were working over the complex numbers, the converse would be true as well: if $V\left(f\right)$ is a cone with vertex at the origin, then the polynomial $f$ must be homogeneous. Why doesn't this work over the reals? Can you find a counterexample?

There's another (closely related) notion of “multiplicity,” namely the multiplicity of intersection of a curve and a line. Given a plane curve $X=V\left(f\right)$ and a line $L$ through a point $p=(a,b)\in X$ , we can define the multiplicity of their intersection as follows: we choose a linear parametrization of $L$ as $\alpha \left(t\right)=(a,b)+t(c,d)$ so that $\alpha \left(0\right)=p$ . The the composition $f\left(\alpha \right(t\left)\right)$ is a polynomial in $t$ with 0 as a root. The intersection multiplicity of $X$ and $L$ at $p$ is defined to be the multiplicity of 0 as a root of $f\left(\alpha \right(t\left)\right)$ , i.e. the power of $t$ in the factorization of $f\left(\alpha \right(t\left)\right)$ .

Suppose that $p$ is a smooth point of $X$ . Then we have

where $F_k(c,d)=f_k(a+c,b+d)$ . The fact that the $F_k(c,d)$ in our expression above are homogeneous polynomials in $c$ and $d$ also helps show that the intersection multiplicity does not depend on our choice of linear parametrization $\alpha$ for $L$ . We see then that the intersection multiplicity of $X$ with $L$ at the smooth point $p$ is 1 except when $c{\left(\frac{\partial f}{\partial x}\right|}_p+d{\left(\frac{\partial f}{\partial y}\right|}_p=0$ , i.e. when $L$ is the tangent line to $X$ at $p$ , in which case the intersection multiplicity is at least 2, and we would need to look at the higher order terms to compute it exactly.

The same computation in the case where $p$ is singular shows that the intersection multiplicity of $X$ with $L$ at $p$ is at least 2 for every line $L$ through $p$ .

Exercises

1. In the previous exercises, you found that the following curves have only one singularity, at $p=(0,0)$ , and calculated the Taylor series expansions at that point. Now, find the multiplicity of each curve at $p$ and find the tangent cone $T{C}_p\left(X\right)$ . This should be a matter of interpreting the TaylorÕs series calculations you havealready made. Sketch the curves and draw in the tangent cones.
   1. $f(x,y)=x^4+y^4-x^2$
   2. $f(x,y)=x^6+y^6-xy$
   3. $f(x,y)=y^2+x^4+y^4-x^3$
   4. $f(x,y)=x^4+y^4-x^{2}y-x{y}^2$
   5. $f(x,y)=x^3+x^2-y^2$
   6. $f(x,y)={(x^2+y^2)}^2-x^2+y^2$
2. Use the same methods to find the singularities, the multiplicity at each singularity, and the tangent cones of the following curves. Since these are a bit more complicated, you will probably want to get a computer to do most of thecalculations. Sketch a graph of the curve and its tangent cone near each singularity. Depending on what program you use, you may have to be careful of the behavior near singular points. Use your information from the tangent cone tointerpret the behavior near singularities.
   1. $f(x,y)=2x^4-3x^{2}y+y^2-2y^3+y^4$
   2. $f(x,y)=2y^2(x^2+y^2)-3y^2-x^2+1$
   3. $f(x,y)=2y^2(x^2+y^2)-2y^2(x+y)-2y^2-x^2+2x+2y$
   4. $f(x,y)={(x^2+y^2)}^3-4x^2{y}^2$
3. One can think of multiplicity as measuring how “bad” a singularity is. We already showed that for a nonsingular point on curve, most lines intersect thatpoint with multiplicity one.
   1. For the curve $f(x,y)=x^3-y^2$ , show that most lines through the origin meet the curve with multiplicity 2.
   2. For the curve $g(x,y)=x^4+2x{y}^2+y^4$ , show that most lines through the origin meet the curve with multiplicity $\ge 3$ .
4. We've mentioned that we ought to be able to make a simple change coordinates so that, for example, a singular point is moved to the origin. The basic idea we were hinting at is that of affine equivalence. An affine change of coordinates is a map of the form
   $$\phi \left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}e\\ f\end{array}\right],\text{where}det\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\ne 0.$$
   We can think of this as basically just a change of variables (but one which is allowed to distort angles and distances). Two curves $f(x,y)$ and $g(x,y)$ are affine equivalent if they differ by an affine change of coordinates $\phi$ . That is, $f(x,y)=g\left(\phi \right(x,y)$ . Show that the curves $f(x,y)=y^2-x^3-x^2$ and $g(x,y)=x^2-2xy-x-y+\frac{1}{4}-y^3$ are affine equivalent.
5. Show that multiplicity is invariant under affine equivalence. That is, if $\phi :C_1\to C_2$ is an affine equivalence, it maps a point with multiplicity $m$ to a point with multiplicity $m$ .
6. This problem is a little different, and its connection to plane curves or algebraic geometry may not be apparent for a while.
   1. What natural numbers $n$ are expressible in the form $n=2x+3y$ where $x$ and $y$ are nonnegative integers? What if we allow $x$ or $y$ to be negative?
   2. What natural numbers $n$ are expressible in the form $n=4x+6y$ where $x$ and $y$ are nonnegative integers? What if we allow $x$ or $y$ to be negative?
   3. What natural numbers $n$ are expressible in the form $n=5x+8y$ where $x$ and $y$ are nonnegative integers? What if we allow $x$ or $y$ to be negative?