Let $X$ be a threefold with isolated ordinary double points $P_1, \dots, P_r$. Let $\pi: Y \to X$ be the blow up of the singular points. Let $E_i$ be the exceptional divisors; these are smooth quadric surfaces $\mathbf{P^1 \times P^1}$.

My question is what is the numerical condition for existence of a projective small resolution in terms of divisors on $Y$?

I see that a necessary condition is existence of a divisor $D$ on $Y$ which restricts "nondiagonally" (that is has bidegree $(a,b)$ for $a \ne b$) on each exceptional divisor. Indeed, otherwise, we have no chance to contract one set of rulings of $E_i$ without contracting the other. Is this also a sufficient condition?

Thank you!

It is known that three-dimensional ordinary double points, that is singular points which complete locally have the equation $xy - zw = 0$ are resolved by a single blow up, with exceptional divisor being a smooth quadric surface $\mathbf{P^1 \times P^1}$. It is also known that sometimes one of the two rulings of the exceptional divisor can be contracted to form a so-called small resolution, that is the one where only a curve is blown down.

So let $X$ be a threefold with isolated ordinary double points $P_1, \dots, P_r$. Let $\pi: Y \to X$ be the blow up of the singular points. Let $E_i$ be the exceptional divisors. I would like to form a small resolution by contracting one of the two rulings on each of the exceptional divisors.

My question is what is the numerical condition for existence of a projective small resolution in terms of divisors on $Y$?

I see that a necessary condition is existence of a divisor $D$ on $Y$ which restricts "nondiagonally" (that is has bidegree $(a,b)$ for $a \ne b$) on each exceptional divisor. Indeed, otherwise, we have no chance to contract one set of rulings of $E_i$ without contracting the other. Is this also a sufficient condition?

Thank you!