# Existence of a projective small resolution

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!

• I believe this is in Mori's paper on extremal threefold contractions. – Jason Starr Mar 29 '19 at 7:03
• Can I please ask which paper of Mori do you mean? – Evgeny Shinder Mar 29 '19 at 8:33
• I am not in my office at the moment. I will look up Mori's paper in my filing cabinet when I can. – Jason Starr Mar 29 '19 at 8:54
• The result of Mori (and Mukai, sorry about forgetting Mukai) is in Section 3, bottom of p. 107 of "On Fano 3-folds with $B_2\geq 2$" by Mori and Mukai. They observe that there exists a birational morphism that contracts one of the two rulings, but not the other, only if the two curve classes of the rulings are not numerically equivalent in the threefold. – Jason Starr Mar 29 '19 at 10:43
• Sorry, my mistake. The paper that Jason refers to is: kurims.kyoto-u.ac.jp/~mukai/paper/Fano1983.pdf (Mori and Mukai have another paper with almost identical name). – Evgeny Shinder Mar 29 '19 at 11:32

The necessary condition that you write is also sufficient. By construction, $$Y$$ is projective. Thus, if there exists any "nondiagonal" divisor class, then (after twisting by a sufficient powers of an ample invertible sheaf) there exists a very ample invertible sheaf $$\mathcal{L}$$ on $$Y$$ whose restriction to each exceptional divisor $$E_i$$ is ample and "nondiagonal". Denote by $$N_i$$ the normal bundle of $$E_i$$ in $$Y$$. This is an antiample invertible sheaf that is diagonal. Thus, there exist integers $$d_i\geq 0$$ such that the invertible sheaf $$\mathcal{M}:=\mathcal{L}\left( \sum_i d_i \underline{E}_i \right),$$ restricts on every $$E_i$$ as a globally generated invertible sheaf that is not big, i.e., it contracts one of the two rulings.

By Michael Artin's theorems in "Algebraization of Formal Moduli, II" (ed.I will add a reference soon) there exists a proper, birational morphism of normal, proper algebraic spaces, $$\nu:Y\to Z,$$ and a nef invertible sheaf $$\mathcal{N}$$ on $$Z$$ such that $$\nu^*\mathcal{N}$$ equals $$\mathcal{M}$$ and such that $$\nu$$ contracts a curve in $$E_i$$ if and only if it has zero intersection number with $$\mathcal{M}$$.

To prove that $$\mathcal{N}$$ is ample, it suffices to use the version of the Nakai-Moishezon Criterion for algebraic spaces proved by János Kollár, Theorem 3.11, p. 248 of the following.

MR1064874 (92e:14008)
Kollár, János
Projectivity of complete moduli.
J. Differential Geom. 32 (1990), no. 1, 235–268.
Nakai-Moishezon hypothesis for curves. To check the hypotheses of the Nakai-Moishezon criterion, begin with irreducible curves. Every irreducible curve in $$Z$$ is the finite image of an irreducible curve in $$Y$$. If that curve is not contained in any divisor $$E_i$$, then its intersection with each of these is nonnegative. Since also the intersection of the curve with $$c_1(\mathcal{L})$$ is positive, the intersection of the irreducible curve with $$c_1(\mathcal{M})$$ is also positive. Also, the restriction of $$\mathcal{M}$$ to each $$E_i$$ is globally generated and has zero intersection number only on those curves contracted by $$\nu$$. Thus, for every irreducible curve in $$E_i$$ that maps finitely to its image in $$Z$$, the intersection number with $$c_1(\mathcal{M})$$ is positive.
Nakai-Moishezon hypothesis for surfaces. Next consider surfaces. As above, it suffices to check for every noncontracted irreducible surface in $$Y$$ that the intersection number with $$c_1(\mathcal{M})\smile c_1(\mathcal{M})$$ is positive. For an irreducible surface that is contained in none of the divisors $$E_i$$, the intersection of every $$E_i$$ with the surface is an effective curve. Also the intersection of the surface with a general member of the linear system of the very ample invertible sheaf $$\mathcal{L}$$ is an effective curve, at least one component of which is contained in no $$E_i$$. Thus, the intersection class of the surface with $$c_1(\mathcal{M})$$ is rationally equivalent to a nontrivial effective curve, at least one component of which is contained in no $$E_i$$. Now apply the previous paragraph to deduce that intersection number of this curve with $$c_1(\mathcal{M})$$ is positive. Since each surface $$E_i$$ is contracted by $$\nu$$, we do not need to check the criterion for these surfaces.
Nakai-Moishezon hypothesis for threefolds. Finally, $$c_1(\mathcal{M})$$ restricts to a divisor class on each $$E_i$$ that is effective. Thus, since the restriction of the effective divisor class $$c_1(\mathcal{M})$$ to a general member of the complete linear system of $$\mathcal{L}$$ is also effective, the intersection class $$c_1(\mathcal{M})\smile c_1(\mathcal{M})$$ is rationally equivalent to an effective curve, not all of whose components are contained in exceptional divisors $$E_i$$. Thus, by the first paragraph once more, the intersection number of this curve with $$c_1(\mathcal{M})$$ is positive, i.e., $$c_1(\mathcal{M})\smile c_1(\mathcal{M})\smile c_1(\mathcal{M})$$ has positive degree. Altogether, this confirms the Nakai-Moishezon criterion for ampleness of $$\mathcal{N}$$ on $$Z$$.