Infinitely many sign-changing and semi-nodal solutions for ...
The group of regular birational maps is...
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The group of regular birational maps is infinitely-generated Shamil Asgarli, Kuan-Wen Lai, Masahiro Nakahara and Susanna Zimmermann Brown University, UMass Amherst, University of Manchester, Université d’Angers Motivation via Finite Fields Consider a birational map f : P 2 99K P 2 defined over a finite field F q . Furthermore, assume that both f and f ≠1 have no F q -points in their indeterminacy loci. In down-to-earth terms, the rational map f is given by f ([x, y, z]) = [f 1 (x, y, z): f 2 (x, y, z): f 3 (x, y, z)] where f 1 ,f 2 ,f 3 are homogenous polynomials of the same degree. The condition that the indeterminacy locus Ind(f ) has no F q - points in P 2 amounts to saying that the polynomials f 1 ,f 2 ,f 3 have no common (nonzero) solution with coordinates in F q . Observation. Such a map f gives rise to a bijection f : P 2 (F q ) æ P 2 (F q ) Question 1. Consider a bijection ‡ : P 2 (F q ) æ P 2 (F q ) which can be interpreted as a permutation ‡ œ S q 2 +q+1 . Does there exist a birational map f : P 2 æ P 2 which induces ‡? Question 2. The set of all such birational maps forms a group BCr 2 (F q ) under composition. Is this group finitely-generated? Introduction The birational automorphisms of P 2 defined over a field k form the Cremona group Cr 2 (k) under composition. Let Ind(f ) µ P 2 denote the set of points where f œ Cr 2 (k) is not defined. We are interested in the subgroup BCr 2 (k) := {f œ Cr 2 (k) : Ind(f )(k) = Ind(f ≠1 )(k)= ÿ}. Elements of BCr 2 (k) are called regular birational maps. In par- ticular, when k = F q is a finite field, every element in BCr 2 (F q ) induces a permutation of the finite set P 2 (F q ). Examples. Elements of PGL 3 (k) are examples of birational maps. Another example is the standard quadratic transformation · : P 2 99K P 2 defined by · ([x : y : z]) = [yz : zx : xy]. Theorem (Noether and Castelnuovo 1901) Suppose k is algebraically closed. Then Cr 2 (k) is generated by PGL 3 (k) and the standard quadratic involution · above. Theorem (Cantat 2012) Suppose k is an arbitrary field. Then Cremona group Cr 2 (k) is infinitely-generated. Theorem (A.-Lai-Nakahara-Zimmermann) Suppose k is a perfect field admitting a quadratic exten- sion of degree of 2. Then the group of regular birational maps BCr 2 (k) is infinitely-generated. Furthermore, BCr 2 (k) is non-normal subgroup of infinite index of Cr 2 (k). Idea behind infinite-generation For each f œ BCr 2 (k), define L f to be the minimal field extension of k such that all the indeterminacy points of f and f ≠1 are defined over L f . If BCr 2 (k) were finitely-generated by f 1 ,f 2 , ..., f r then for each f œ BCr 2 (k), L f would be contained in the compositum of L f1 , ..., L fr , and in particular [L f : k] Æ [L f1 : k] · [L f2 : k] ··· [L fr : k] which would imply that the set {[L f : k]: f œ BCr 2 (k)} is bounded. Therefore, it is enough to prove that, [L f : k] achieves arbitrarily large values as f varies in BCr 2 (k). Constructing the Cremona map: Algebra For each n Ø 2, pick n +4 points on the plane where the 4 points and n points separately form two Galois orbits. Consider the linear system of degree 2n +1 curves passing through the 4 points with multiplicity n, and through the n points with multiplicity 2. See Figure 1 for the case n =6. Assuming the points are in sufficiently general position, this vector space has dimension Q c c c c c c c c c a 2n +3 2 R d d d d d d d d d b ≠ 4 Q c c c c c c c c a n(n + 1) 2 R d d d d d d d d b ≠ 3n =3 since each multiplicity k point imposes k(k + 1)/2 linear condi- tions. Choose a basis {f 1 ,f 2 ,f 3 } for this vector space, and define a rational map f : P 2 æ P 2 such that f (x, y, z)=[f 1 (x, y, z): f 2 (x, y, z): f 3 (x, y, z)] One can check that f œ BCr 2 (k), and the indeterminacy locus consists of exactly n +4 points above. Moreover, [L f : k] Ø n and we seem to be done! Subtlety: How can we choose the points so they are in general position and form Galois orbits? Figure 1: Degree 13 curve passing through 10 points (n =6) Constructing the Cremona map: Geometry There is a geometric construction for the Cremona map discussed in the previous section. Here is a key diagram: X C C Õ !! P 2 f // P 2 . The spaces C and C Õ will be both conic bundles. Let T be a quadratic extension of k. Pick four points {a 1 ,a 2 ,b 1 ,b 2 } in P 2 (T ) such that {a 1 ,a 2 } and {b 1 ,b 2 } form Gal(T/k)-orbits, and no three of these four points are collinear. Step 1. Blow-up P 2 along {a 1 ,a 2 ,b 1 ,b 2 } to obtain a conic bundle C æ P 1 , which is fibered in the conics passing through {a 1 ,a 2 ,b 1 ,b 2 }. Let ¸ 1 µ P 2 be a line over T passing through a 1 , but not a 2 ,b 1 ,b 2 , and let ¸ 2 be its Gal(T/k)-conjugate. Step 2. There exists a closed point x œ ¸ 1 defined over an even degree extension K/k but not over any proper subfield, such that r of the Gal(K/k)-conjugates of x lie on ¸ 1 (resp. ¸ 2 ), where r = 1 2 [K : k] and such that all n =2r conjugates of x lie in the distinct fibers of C æ P 1 . Step 3. Blow-up C at these n points to get a map X æ C. Step 4. Blow-down the strict transforms of the n blown-up points to get C Õ . Check that the resulting surface C Õ is smooth and admits a conic fibration. Moreover, C Õ is also a blow-up of P 2 at four points. References [Can09] Serge Cantat, Birational permutations, C. R. Math. Acad. Sci. Paris 347 (2009), no. 21-22, 1289–1294. MR 2561040 The first author has been supported by the NSF Grant DMS-1701659.
Transcript of The group of regular birational maps is...
The group of regular birational maps is infinitely-generated
Shamil Asgarli, Kuan-Wen Lai, Masahiro Nakahara and Susanna Zimmermann
Brown University, UMass Amherst, University of Manchester, Université d’Angers
Motivation via Finite Fields
Consider a birational map f : P2 99K P2 defined over a finitefield F
q
. Furthermore, assume that both f and f
≠1 have noF
q
-points in their indeterminacy loci. In down-to-earth terms,the rational map f is given by
f ([x, y, z]) = [f1(x, y, z) : f2(x, y, z) : f3(x, y, z)]where f1, f2, f3 are homogenous polynomials of the same degree.The condition that the indeterminacy locus Ind(f ) has no F
q
-points in P2 amounts to saying that the polynomials f1, f2, f3have no common (nonzero) solution with coordinates in F
q
.Observation. Such a map f gives rise to a bijection
f : P2(Fq
) æ P2(Fq
)Question 1. Consider a bijection ‡ : P2(F
q
) æ P2(Fq
) whichcan be interpreted as a permutation ‡ œ S
q
2+q+1. Does thereexist a birational map f : P2 æ P2 which induces ‡?Question 2. The set of all such birational maps forms a groupBCr2(Fq
) under composition. Is this group finitely-generated?
Introduction
The birational automorphisms of P2 defined over a field k formthe Cremona group Cr2(k) under composition. Let Ind(f ) µ P2
denote the set of points where f œ Cr2(k) is not defined. We areinterested in the subgroup
BCr2(k) := {f œ Cr2(k) : Ind(f )(k) = Ind(f≠1)(k) = ÿ}.
Elements of BCr2(k) are called regular birational maps. In par-ticular, when k = F
q
is a finite field, every element in BCr2(Fq
)induces a permutation of the finite set P2(F
q
).Examples. Elements of PGL3(k) are examples of birationalmaps. Another example is the standard quadratic transformation· : P2 99K P2 defined by · ([x : y : z]) = [yz : zx : xy].
Theorem (Noether and Castelnuovo 1901)
Suppose k is algebraically closed. Then Cr2(k) is generated by
PGL3(k) and the standard quadratic involution · above.
Theorem (Cantat 2012)
Suppose k is an arbitrary field. Then Cremona group Cr2(k)is infinitely-generated.
Theorem (A.-Lai-Nakahara-Zimmermann)Suppose k is a perfect field admitting a quadratic exten-
sion of degree of 2. Then the group of regular birational
maps BCr2(k) is infinitely-generated. Furthermore, BCr2(k)is non-normal subgroup of infinite index of Cr2(k).
Idea behind infinite-generation
For each f œ BCr2(k), define L
f
to be the minimal field extensionof k such that all the indeterminacy points of f and f
≠1 are definedover L
f
. If BCr2(k) were finitely-generated by f1, f2, ..., f
r
thenfor each f œ BCr2(k), L
f
would be contained in the compositumof L
f1, ..., L
f
r
, and in particular[L
f
: k] Æ [Lf1 : k] · [L
f2 : k] · · · [Lf
r
: k]which would imply that the set
{[Lf
: k] : f œ BCr2(k)}is bounded. Therefore, it is enough to prove that, [L
f
: k] achievesarbitrarily large values as f varies in BCr2(k).
Constructing the Cremona map: Algebra
For each n Ø 2, pick n + 4 points on the plane where the 4 pointsand n points separately form two Galois orbits. Consider the linearsystem of degree 2n + 1 curves passing through the 4 points withmultiplicity n, and through the n points with multiplicity 2. SeeFigure 1 for the case n = 6. Assuming the points are in su�cientlygeneral position, this vector space has dimension
Q
ccccccccca
2n + 32
R
dddddddddb
≠ 4Q
cccccccca
n(n + 1)2
R
ddddddddb≠ 3n = 3
since each multiplicity k point imposes k(k + 1)/2 linear condi-tions. Choose a basis {f1, f2, f3} for this vector space, and definea rational map f : P2 æ P2 such that
f (x, y, z) = [f1(x, y, z) : f2(x, y, z) : f3(x, y, z)]One can check that f œ BCr2(k), and the indeterminacy locusconsists of exactly n +4 points above. Moreover, [L
f
: k] Ø n andwe seem to be done!
Subtlety: How can we choose the points so they are in generalposition and form Galois orbits?
Figure 1: Degree 13 curve passing through 10 points (n = 6)
Constructing the Cremona map: Geometry
There is a geometric construction for the Cremona map discussedin the previous section. Here is a key diagram:
X
��
C��
C Õ!!
P2 f
//P2.
The spaces C and C Õ will be both conic bundles.Let T be a quadratic extension of k. Pick four points {a1, a2, b1, b2}in P2(T ) such that {a1, a2} and {b1, b2} form Gal(T/k)-orbits, andno three of these four points are collinear.Step 1. Blow-up P2 along {a1, a2, b1, b2} to obtain a conicbundle C æ P1, which is fibered in the conics passing through{a1, a2, b1, b2}. Let ¸1 µ P2 be a line over T passing through a1,but not a2, b1, b2, and let ¸2 be its Gal(T/k)-conjugate.Step 2. There exists a closed point x œ ¸1 defined over an evendegree extension K/k but not over any proper subfield, such thatr of the Gal(K/k)-conjugates of x lie on ¸1 (resp. ¸2), wherer = 1
2[K : k] and such that all n = 2r conjugates of x lie in thedistinct fibers of C æ P1.Step 3. Blow-up C at these n points to get a map X æ C.Step 4. Blow-down the strict transforms of the n blown-up pointsto get C Õ. Check that the resulting surface C Õ is smooth and admitsa conic fibration. Moreover, C Õ is also a blow-up of P2 at four points.
References
[Can09] Serge Cantat, Birational permutations, C. R. Math. Acad. Sci. Paris347 (2009), no. 21-22, 1289–1294. MR 2561040
The first author has been supported by the NSF Grant DMS-1701659.
