Singularities of maps and birational rigidity
Tommaso de Fernex“Singularity Theory and Commutative Algebra” – Japan, September 2005
This talk will be devoted to an overview of the methods involved in the study ofbirational maps between a certain class of algebraic varieties, called Mori fiber spaces. I
will explain how the theory of singularities of pairs naturally comes into the picture. Allvarieties will be defined over the complex numbers.
A Mori fiber space is by definition a normal projective variety with terminal, Qfactorial singularities, endowed with an extremal Mori contraction of fiber type.1 Thesevarieties are one of the two expected “final output” of the Minimal Model Program, theother one consisting of minimal models. As one would like to choose such a variety as the“simplest” object to represent its birational class, it is then natural to inquire about theunicity of such a choice. This led to the notion of birational rigidity.
We say that a Mori fiber space V is birationally superrigid if the only birational
maps from V to another Mori fiber space are the automorphisms of V . This is a verystrong condition, which in particular implies that V is not rational, V is the only Morifiber space in its birational class, and Bir(V ) = Aut(V ). A slightly weaker notion is thatof birational rigidity, for which, roughly speaking, the Mori fiber structure of V is required
to be “birationally” defined in a unique way.
The first example of birational superrigidity was discovered by Iskovskikh andManin, with their proof of non-rationality of smooth quartic threefolds in P4 . This also
gave a counter-example to a famous problem posed by Lüroth.2 The method introducedin the proof, known as the method of maximal singularities, extends the original ideas ofNoether and Fano to effectively bound the singularities of a birational map.
During the talk I will mainly focus on the case of projective Fano hypersurfaces,
discussing how the result of Iskovskikh and Manin generalizes to higher dimensions. Considering the problem of birational rigidity for a hypersurface V ⊂ PN , it becomes quickly
clear that the question is pertinent only when the degree of V is N . The general conjecture,
posed by Pukhlikov, is that every smooth hypersurface of degree N in PN is birationallysuperrigid if N ≥ 4. In a series of publications, starting with Iskovskikh and Manin’scontribution, then Pukhlikov’s, Cheltsov’s, and finally, a paper that I have coauthoredwith Ein and Mustaţǎ, this conjecture has been progressively established in a few low dimensional cases. Pukhlikov also found a proof in all dimensions under certain conditions
on the equation defining the hypersurface. All together, this is what we know so far:
Theorem. Let V be a smooth hypersurface of degree N in PN . If 4 ≤ N ≤ 12, then V
is birationally superrigid. Moreover, if V is chosen to be sufficiently general in its moduli
space, then superrigidity holds for every N ≥ 4.
It is interesting to observe that the non-rationality for general V was already known
by a result of Kollár, but even this weaker property is unknown to hold for every V .
1Fano manifolds with Picard number 1 are, in a trivial way, examples of Mori fiber spaces.
2Another counter-example to the Lüroth problem was independently found by Clemens and Griffiths,

and more were found shortly afterwards by Artin and Mumford.
1Let me now give here a very brief outline of the general method.3 The basic idea isto assume the existence of a birational map
φ : V 99K V 0 ,φ 6= isomorphism,

between V and some other Mori fiber space V 0 . Then one would like to quantify somehow
the singularities of the indeterminacy locus of the map, hoping to get a contradiction byconsidering the degrees of the variety and the equations defining the map. The following
is the first key fact.
Noether-Fano Ineqality. If B ⊂ V is the subscheme cut by the equations defining V ,and r is the degree of such equations, then the pair (V, 1r B) is not canonical.
The difficulty is to translate the information contained in this statement in termsof more classical notions of singularities that can be more easily compared to the variousdegrees. Following an idea of Corti, one can tackle this problem in two steps, by reducing
first to work with log-canonical thresholds.
Corollary of Shokurov’s Connectedness Principle. If a point p is a center of noncanonicity for a pair (V, λB) and X ⊂ V is a general hyperplane section through p, thenthe same point is a center of non log-canonicity for the pair (X, λB|X ).
The second step is to compare log-canonical thresholds with multiplicities. The
following was proven by Corti in dimension 2 and then generalized in all dimensions byEin, Mustaţǎ and myself.
Theorem. Let R = OX,p be a n-dimensional regular local ring, and assume that a ⊂ R isa zero dimensional ideal. Then, denoting by e(a) the Samuel multiplicity of a and by lc(a)the log-canonical threshold of a, we havee(a) ≥ (n/ lc(a))n .
Finally the circle is complete: we started with a geometric information on V (theexistence of a certain rational map) and ended up with a new geometric information on
V (the existence of some subscheme with very high multiplicity at some point).
However, although the above inequality suffices to draw the conclusion in the caseof quartic threefolds, the gap with the degree of V becomes too large as the dimension
increases. A projection method, initiated by Pukhlikov and then completed in a joint
work by Ein, Mustaţǎ and myself, leads to the proof up to the case N = 12. The main
ingredient is the following inequality on log-canonical thresholds.
Theorem. Let f : X → Y be a smooth proper morphism of smooth algebraic varieties,and let Z ⊂ X be a locally complete intersection subscheme, of pure codimension k, suchthat dim Z = dim Y − 1 and f |Z is finite. Thenlc(Y, f∗ [Z]) ≤ (lc(X, Z)/k)k .
At the moment, as N gets larger than 12, one runs into some technical difficulty incontrolling the singularities after projection.

3The various notions of singularities that appear in this abstract will be introduced during the talk.
2