This note will be an elementary introductory intersection theory.
1. Extending the notion of degree of curves on $\mathbb{P}^n$. Let $X$ be a smooth curve on $\mathbb{P}^2$, then $X$ is of dimension 1, and it is represented by the zero locus of a non-constant homogeneous polynomial $f$ in $k[x_0,x_1,x_2]$. We then define the degree of $X$ as the degree of $f$. But...on $\mathbb{P}^3$, a curve is not as simple as a curve in $\mathbb{P}^2$. For example, we consider the twisted cubic curve $X:=\{[x^3:x^2y:xy^2:y^3]\text{ where }[x:y]\in\mathbb{P^1}\}\subset \mathbb{P}^3$ (as we will see later, $X\cong\mathbb{P}^1$). If we denote the coordinate of $\mathbb{P}^3$ as $[x_0:x_1:x_2:x_3]$, then this set is the zeros locus of
$$x_0x_3 = x_1x_2, x_0x_2=x_1^2, x_1x_3=x_2^2$$
That means $X$ is the zero locus of a polynomial system, other than a particular polynomial. One may want to prove that $X\ne Z(f)$, for some non-constant homogeneous polynomial in $k[x_0,x_1,x_2,x_3]$. Note that $\mathbb{P}^3$ has dimension 3, and by the Krull's principal ideal theorem, the dimension of $Z(f)$ is 2, which is strict larger than the dimension of a curve. Hence, to obtain a curve in $\mathbb{P}^3$, we should cut it out by at least two polynomials. But then the difficulty arises, how one can define the degree of such curves on $\mathbb{P}^3$?
An interesting way to do this is to use intersection theory. Recall that if $X$ is a smooth projective plane curve is given by the zero locus of a homogeneous polynomial $f\in k[x_0, x_1, x_2]$, and $L$ is any line in $\mathbb{P}^2$, such that $L\not\subset X$, then by Berzout's theorem, we know that $\deg X$ is the number of point that $X$ intersect $L$ counting with multiplicities. And we will define the degree of a curve $X\subset\mathbb{P}^n$ by counting the number of intersection point between $X$ and a hyperplane $H$, counting with multiplicity. To do this, we need divisors, and of course, Riemann-Roch's theorem, but we will not discuss about it much in this post.
2. Divisors, intersection divisors, and hyperplane divisors. We fix $X$ an irreducible smooth curve in $\mathbb{P}^n$. A divisor $D$ on $X$ is a formal sum $D:=\sum_{P\in X}n_PP$, where $n_P\in X$, and only finitely many $n_P\ne 0$. The set of all divisors on $X$ is denoted $Div(X)$. It can be seen that $Div(X)$ has a natural group structure induced from $\mathbb{Z}$. If $D:=\sum_{P\in X}n_PP\in Div(X)$, we define $D(p):=n_P$, and $\deg(D):=\sum_{P\in X}n_P$. It can be easily seen that the degree map is a group homomorphism from $Pic(X)$ to $\mathbb{Z}$.
Let us denote $k(X)$ the function field of $X$, consist of all rational functions of the form $\frac{f}{g}$, where $g\ne 0$, and $f,g\in k[x_0,...,x_n]/I(X)$ are homogeneous of the same degree. And $k(X)$ is called the field of rational functions on $X$. The stalk at a point $p\in X$ is a DVR, that means, there exists a local coordinate $\pi_P$ at $P$ such that $\pi_P(P)=0$, and $\pi_P$ is a generator of the maximal ideal of the local ring at $P$, and there exists an open neighborhood $U_p$ at $p$ such that $\pi_P(x)\ne 0, \forall x\in U_p\setminus\{P\}$. Also, for any $F$ in $k(X)$ defined at $P$, locally at $P$, there exists only one $n\ge 0$ such that $F=\pi_P^nG$, where $G(P)\ne 0$, and such $n$ is called the order of $F$ at $P$, denoted by $ord_P(F)$.
By this, for any $f\in k[x_0,...,x_n]$ is a homogeneous polynomial of degree $d$, assume that at $P$, the coordinate $x_0\ne 0$, then $\frac{f}{x_0^d}\in K(X)$ defined at $P$, and locally at $p$, there exists only one $n:=ord_P(\frac{f}{x_0^d})$ such that $\frac{f}{x_0^d}=\pi_P^nF$, where $F(p)\ne 0$. Because $x_0(P)\ne 0$, $x_0^d$ does not affect to the order of $f$ at $p$. Hence, we can define $ord_P(f):=ord_P(\frac{f}{x_0^d})$. And the intersection divisor of $f$ is defined as follows
$$div(f):=\sum_{P\in X}ord_P(f)P$$. If $f$ is of degree 1, then $f$ is called the hyperplane divisors. We often denote the hyperplane divisor as $H$. If $F=\frac{f}{g}\in k(X)$, then the principal divisor is defined as $div(F):=div(f)-div(g)$, i.e. it is the difference of the two intersection divisors.
The name "intersection divisor" can be expressed as follows. Assume that $P\in V(f)\cap X$, then for sure $ord_P(f)\ge 1$. Conversely, if $ord_P(f)\ge 1$, then $P\in V(f)\cap X$. We are now ready to define the degree of $X$.
Definition 2.1. Let $X\subset\mathbb{P}^n$ be an irreducible smooth projective curve, and $H$ be any hyperplane divisor, such that $X\not\subset H$, then the degree of $X$ is $\deg(X)=\deg(H)$.
We should check first the degree of $X$ is well-defined, i.e. for any $H'$ is another hyperplane divisor then $\deg H=\deg H'$. It directly follows by the following two propositions below.
Proposition 2.2. Let $X$ be defined as above, and $f\in k(X)$, then $\deg(div(f))=0$.
Proof Sketch. One way to prove it is to use the the Riemann-Roch's theorem (with Serre's duality-in this case, look at my previous note for Serre's duality). We will present another way to see this, and this solution will play an important role for our later notes. If $X, Y$ are smooth projective curves, and $\phi:X\to Y$ is a surjective morphism, then for all $q\in Y$, and $p\in \phi^{-1}(q)$. Let $\pi_q, \pi_p$ be the local coordinates at $q,p$, respectively. Then the ramification index $e_p$ is defined as the highest multiplicity $\pi_q\circ\phi=\pi_p^{e_p}g$ (which is the pull-back $\phi^*(\pi_q))$, for $g(p)\ne 0$ in an open neighborhood of $p$. And very similar to what we discussed on algebraic number theory, for all point $q\in Y$, $\sum_{\phi(p)=q}e_p$ is a constant, which is called the degree of the morphism $\phi$. The sum $\sum_{\phi(p)=q}e_pp\in Div(X)$ is called the pull-back of $q\in Div(Y)$, and denoted by $\phi^*(q)$. It defines a group homomorphism from $Div(Y)$ to $Div(X)$.
By using this, if we can represent $div(f)=\sum_{n_P>0}n_PP-\sum_{n_Q>0}n_QQ$, where $P$'s and $Q$'s are zeros and poles of $f$, respectively. Assume that $f:=\frac{f_1}{f_2}\in k(X)$, then the map $\phi: X\to \mathbb{P}^1$ defined by sending $p\in X$ to $[f_1(p):f_2(p)]$ is a morphism (at this point, you may wonder what happens if $p$ is the common-zero of $f_1$ and $f_2$, but we will discuss it in later post, where we will do "analytic continuation" in algebraic geometry). And hence, the zero part $\sum_{n_P>0}n_P P$ is just the pull back of $0$, i.e. $\sum_{n_P>0}n_P P=\phi^*(0)$. And the pull back of $\infty$ is exactly $\sum_{n_Q>0}n_Q Q$. By our previous remark, $\sum_{n_P}n_P=\sum_{n_Q}n_Q$ is a constant. Hence, $\deg(div(f))=0$.
(Q.E.D)
Via such difficult proof, one can easily obtain
Proposition 2.3. Let $F_1, F_2$ be two homogeneous polynomials of the same degree, and $X$ is defined as above, such that $X\not\subset V(F_i)(i=1,2)$. Then $\deg(div(F_1))=\deg(div(F_2))$.
Proof. Note that $F:=\frac{F_1}{F_2}\in K(X)$, by Proposition 2.2, $\deg(div(F))=\deg(div(F_1))-\deg(div(F_2))=0$, i.e. $\deg(div(F_1))=\deg(div(F_2))$. (Q.E.D)
From the Proposition 2.3, one can see that the intersection divisor just depends on the degree of the polynomial. And hence, the degree of $X$ is well-defined. In the case $X$ is plane curve given by the zero locus of a polynomial $F$, i.e. $X\subset \mathbb{P}^2$, by Berzout's theorem, we know that $\deg(F)=\deg(div(H))=\deg(X)$. And the following proposition is actually the generalization of Berzout's theorem.
Proposition 2.4. Let $X$ be defined as above, and $F$ is a homogeneous polynomial of degree $d$ in $k[x_0,...,x_n]$ that does not vanish identically on $X$ (i.e. $X\not \subset V(F))$, then $\deg(div(F))=\deg X.\deg F$.
Proof. It can be seen by Proposition 2.3 that $\deg(div(X))$ just depends on the degree of $F$, and we can replace $F$ by $H^d$, where $H$ is a hyperplane, that does not contain $X$. By our definition, we have $\deg(div(H^d))=d\deg(H)=d\deg(X)$. (Q.E.D)
As a corollary, we have
Corollary 2.5. Let $F,X$ be defined as in Proposition 2.4, then the $\#\{X\cap V(f)\}\le \deg(X)\deg(F)$. And the equality occurs when we count multiplicities.
At the end, you can see we have successfully extend the notion of degree for smooth projective curve in $\mathbb{P}^n$. But we have not given any example, and the reader may ask, what is the degree of the curve we mentioned at the first section? I will come back with this example when we discuss more details about morphism to $\mathbb{P}^n$ associated with a divisor, and the pull-back of hyperplane divisor.
No comments:
Post a Comment