Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK\_X = \mathcal{O}\_X(b)$. 


Furthermore, if $f:Y\rightarrow X$ is the blow-up of the vertex then $Y$ is smooth and we may write $K\_Y = f^{\*}K\_{X}+dE$, where $E$ is the exceptional divisor, and $d$ is a rational number.


How can I determine the numbers $a,b,d$ ?



Cone over the Veronese surface
Let me start being a little nitpicking with the formulation of the question. The fact that $X$ is $\mathbb Q$-factorial does not in itself imply that such $a$ and $b$ exists. One also needs the fact that the Picard number of $X$ is $1$. This is indeed true, but perhaps should be mentioned. In fact, the Picard group of $X$ is $\mathbb Z$ which was silently used in some answers. This is especially not obvious in cases where quotients are taken. Here it is (of course?) trivial, but in my opinion deserves mentioning. One possible way to see it is that $X\setminus \{P\}$ (where $P$ is the vertex) is an $\mathbb A^1$-bundle over $\mathbb P^2$, so its Picard group is $\mathbb Z$ and hence the group of Weil divisors on $X$ is also $\mathbb Z$ (this makes sense as $X$ is normal, because the Veronese embedding is projectively normal). This, by the way, is also a proof that $X$ is $\mathbb Q$-factorial (since it is projective), and that the Picard group is a subgroup which is also isomorphic to $\mathbb Z$.


So, let's get to answering the questions:


First, let us determine $a$ and $b$. These do not require blowing up. Let $H\subset X$ denote a general hyperplane section of $X$, so $H\simeq V\simeq \mathbb P^2$. Then since $V\subset \mathbb P^2$ is the $2$-uple embedding, $H\left|\_H\right.$ is (linearly equivalent to the image of) a conic in $\mathbb P^2$. By adjunction $$-3=\deg K\_H = \deg \left((K\_X+H)\left|\_H \right.\right)$$ (notice that $H$ avoids the singular point of $X$, so this is OK). It follows that $$\deg K\_X\left|\_H\right.= -5.$$


Since the Picard group of $X$ is $\mathbb Z$ (!!) it follows that $2K\_X\sim -5H$, i.e., that $a=2$ and $b=-5$ (or any multiple of these values, e.g., $a=-2$ and $b=5$ works as well).


Next, let us determine $d$: From the equation $K\_Y\sim f^\*K\_X + dE$, using adjunction, one obtains that $$(d+1)E\left|\_E\right.\sim K\_E.\tag{$\*$}$$
Since $f$ was obtained as a blow-up of $\mathbb P^6$ at a point, $-E\left|\_E\right.$ is linearly equivalent to the restriction of the hyperplane class of the exceptional $\mathbb P^5$ and hence it is a conic on $E$. Combined with $(\*)$ this implies that $$-2(d+1)=-3,$$
that is, that $$d=\frac 12.$$


