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!

  • $\begingroup$ I believe this is in Mori's paper on extremal threefold contractions. $\endgroup$ Mar 29, 2019 at 7:03
  • $\begingroup$ Can I please ask which paper of Mori do you mean? $\endgroup$ Mar 29, 2019 at 8:33
  • $\begingroup$ I am not in my office at the moment. I will look up Mori's paper in my filing cabinet when I can. $\endgroup$ Mar 29, 2019 at 8:54
  • $\begingroup$ 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. $\endgroup$ Mar 29, 2019 at 10:43
  • 1
    $\begingroup$ 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). $\endgroup$ Mar 29, 2019 at 11:32

1 Answer 1


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 $Z$.

  • $\begingroup$ Dear Jason, thank you very much for this extended explanation. $\endgroup$ Mar 29, 2019 at 11:41
  • $\begingroup$ You are welcome. The article by Artin might actually be "Algebraic construction of Brieskorn's resolutions", not "Algebraization of formal moduli, II" . . . $\endgroup$ Mar 29, 2019 at 11:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.