Hartshorne Exercise V.3.3.
Let $\pi:\tilde X\ra X$ be a monoidal transformation, and let $D$ be a very ample divisor on $X$. Show that $2\pi^*D-E$ is ample on $\tilde X$. [Hint: Use a suitable generalization of (I, Ex. 7.5) to curves in $\PP{k}{n}$.]
We apply Nakai Moishizon criterion. One sees easily that,
$$(2\pi^*D-E)^2=4D^2-1\ge3,$$
and
$$(2\pi^*D-E).E=1.$$
Any non-exceptional curve in $\tilde X$ is the strict transform of its image, so it remains to show that,
$$(2\pi^*D-E).\tilde C=(2\pi^*D-E).(\pi^*C-rE)=2D.C-r>0,$$
where $C\subset X$ is any curve and $r$ is the multiplicity of $C$ at the blowup point, $P$. Assume that $P$ is a singular point of $C$, else there is nothing to show. $D$ gives a projective embedding of $X\ra\PP{k}{n}$, and the corresponding embedding of $C$ has degree $D.C$. So, we must show that for a surface $X\subset\PP{k}{n}$ and a curve $C\subset X$, $\mu_P(C)\le \deg C$ for every $P\in C$. As usual, we may assume $D$ is smooth and irreducible by Bertini’s theorem. In Problem 11 (V, Ex. 3.2), we saw that $\tilde C.\tilde D=C.D-r\mu_P(D)=C.D-r$ so
$$2C.D=2\tilde C.\tilde D+2r\mu_P(D)=\tilde C.(2\pi^*D-E)-\tilde C.E+2r\ge 2r+1-\tilde C.E=r+1.$$
Therefore, $2C.D-r\ge 1$, as desired.
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