2 added some dollar signs, and hopefully didn't introduce any errors; added 2 characters in body

OK, here is another way to see it more in line with what you had in mind I think. Write your L $L$ as OO(D) $\mathcal O(D)$ for some divisor D $D$ on X. $X$. Set J_1 $J_1$ to be the ideal sheaf defined by OO(-D$\mathcal O(-D) \cap OO_X \mathcal O_X$ and J_2 $J_2$ to be the ideal sheaf defined by OO(D$\mathcal O(D) \cap OO_X \mathcal O_X$ (intersections taken inside of K_X). $K_X$). Let Y_i $Y_i$ be the closed subschemes of X $X$ defined by these ideal sheaves (they have dimension smaller than that of X). $X$). Then we have the exact sequences

0 ->

$$0 \to J_1(kD) -> OO(kD\to \mathcal O(kD) -> OO_Y1(kD\to \mathcal O_{Y_1}(kD) -> 0 0 -> \to 0$$

$$0 \to J_2((k-1)D) -> OO((k-1)D\to \mathcal O((k-1)D) -> OO_Y2((k-1)D\to \mathcal O_{Y_2}((k-1)D) -> 0\to 0$$

The two left hand terms are equal by construction. Then by the induction hypothesis, and chasing the Euler characteristics, \chi(kD) $\chi(kD) - \chi((k-1)D) chi((k-1)D)$ is a numerical polynomial. This implies that that \chi(kD) $\chi(kD)$ itself is a numerical polynomial (Section 1.7 of Harshorne's Algebraic Geometry).

(Here I swept something under the rug, because the subschemes Y_i $Y_i$ may not be as nice as X $X$ was. But they are at least proper, and we should show that the result we want is that for a proper variety W, \chi(kD) $W$, $\chi(kD)$ is polynomial for a divisor D. $D$. Then reduce this to the case where W $W$ is reduced by looking at the inclusion of W_red $W_\mathrm{red}$ into W. $W$. Then further reduce to the case where W $W$ is integral.)

1

OK, here is another way to see it more in line with what you had in mind I think. Write your L as OO(D) for some divisor D on X. Set J_1 to be the ideal sheaf defined by OO(-D) \cap OO_X and J_2 to be the ideal sheaf defined by OO(D) \cap OO_X (intersections taken inside of K_X). Let Y_i be the closed subschemes of X defined by these ideal sheaves (they have dimension smaller than that of X). Then we have the exact sequences

0 -> J_1(kD) -> OO(kD) -> OO_Y1(kD) -> 0

0 -> J_2((k-1)D) -> OO((k-1)D) -> OO_Y2((k-1)D) -> 0

The two left hand terms are equal by construction. Then by the induction hypothesis, and chasing the Euler characteristics, \chi(kD) - \chi((k-1)D) is a numerical polynomial. This implies that that \chi(kD) itself is a numerical polynomial (Section 1.7 of Harshorne's Algebraic Geometry).

(Here I swept something under the rug, because the subschemes Y_i may not be as nice as X was. But they are at least proper, and we should show that the result we want is that for a proper variety W, \chi(kD) is polynomial for a divisor D. Then reduce this to the case where W is reduced by looking at the inclusion of W_red into W. Then further reduce to the case where W is integral.)