Reduced divisors, multiplicity at a point, and pullback of divisors

(Prime and reduced divisors)

On a Noetherian integral separated scheme $X$ regular in codimension one, a prime divisor is a closed integral subscheme of codimension one. A Weil divisor $D = \sum_i n_i D_i$ with the $D_i$ distinct prime divisors is effective if all $n_i \geq 0$, and reduced if every $n_i \in \{0, 1\}$.

Integrality of $X$ gives the function field $K(X)$. Regularity in codimension one makes the local ring $\OO_{X,\eta_Y}$ at the generic point of each prime divisor $Y$ a discrete valuation ring, so every $f\in K(X)^\times$ has an order $v_Y(f)$ along $Y$, which defines $\operatorname{div}(f)$ (D-5PQ5W).

(Multiplicity at a point)

Let $X$ be a smooth variety and $D$ an effective divisor with local equation $f \in \OO_{X,p}$ at a point $p$. The multiplicity of $D$ at $p$ is the largest $m$ with $f \in \mfm_p^m$. It is $0$ when $p \notin \supp D$, and $1$ exactly when $D$ is smooth at $p$.

(Pullback)

Let $\phi \colon X \to Y$ be a morphism of integral schemes and $D$ a Cartier divisor on $Y$ with local equations $\{(U_i, f_i)\}$, such that $\phi(X) \not\subseteq \supp D$. The pullback $\phi^* D$ is the Cartier divisor on $X$ with local equations $\{(\phi^{-1} U_i, f_i \circ \phi)\}$. It satisfies $\OO_X(\phi^* D) \cong \phi^* \OO_Y(D)$, and it respects linear equivalence, so it induces $\phi^* \colon \Pic Y \to \Pic X$.

For the plane curve $D = V(y^2 - x^3)$ in $\AA^2$, the multiplicity at the origin is $2$, since $y^2 - x^3 \in \mfm^2 \setminus \mfm^3$, and the multiplicity is $1$ at every other point of $D$. For the double cover $\phi \colon \AA^1 \to \AA^1$, $t \mapsto t^2$, the pullback of the divisor $[0]$ is $2[0]$.

What is a prime divisor? A reduced divisor?

For a prime divisor $D$ and $p \in \supp D$, what is the multiplicity of $D$ at $p$?

What is the pullback of a divisor?

Which hypotheses on a scheme define Weil divisors and the principal divisor of a rational function?

Dependencies

D-5PQ5WWeil and Cartier divisors, and when they agreeuses

What links to this

Weil and Cartier