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Suppose $X$ is a non-singular toric variety that is the closure of the complex torus $X^\circ \simeq T_{\mathbb C}$. Let $Y_1,\dots, Y_N$ be the closures of codimension one $T_{\mathbb C}$-orbits, so that $X$ is the union $X^\circ \cup Y_1 \cup \dots \cup Y_N$.

Let $u: \mathbb P^1 \to X$ be a holomorphic curve that intersects $X^\circ$, and therefore intersects each of the divisors $Y_1,\dots,Y_N$ at isolated points. I would like to see a proof of the fact that the first Chern number $c_1(u^*TX)$ is the sum of the intersection numbers of $u$ with the divisors $Y_1,\dots,Y_N$.

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    $\begingroup$ The divisor $\sum Y_i$ is an anticanonical divisor of $X$. This is a standard fact about toric varieties, see e.g. Fulton Introduction to Toric Varieties, 4.3. $\endgroup$
    – abx
    Feb 16, 2022 at 19:57

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