Density of rational points on
Del Pezzo surfaces of degree one
Ronald van Luijk, Jelle Bulthuis Oaxaca
Surface: smooth, projective, geometrically integral scheme of finite type over a field, of dimension 2.
SurfaceX is Del Pezzoif anticanonical divisor−KX is ample.
Degreeof Del Pezzo surfaceX is intersection numberd = KX · KX.
Examples:
I Intersection of two quadrics in P4 (−KS very ample; d = 4).
I A smooth cubic surface inP3 (−KS very ample; d = 3).
I Smooth double cover of P2, ramified over a quartic (d = 2). (Anticanonical map is the projection to P2).
Theorem(Segre, Manin, Koll´ar, Pieropan, Salgado–Testa–V´arilly-Alvarado, Festi-vL).
LetS be a Del Pezzo surface of degree d ≥ 2over a field k. SupposeP ∈ S (k) is a rational point. Ifd = 2and k is infinite, then suppose, furthermore, thatP does not lie on four exceptional curves, nor on the ramification locus of the anticanonical map. ThenS is unirational over k.
Theorem(Koll´ar, Mella)
LetS be a Del Pezzo surface of degree d = 1over a field k of characteristic not equal to2. If S admits a(certain?) conic bundle structure, thenS is unirational.
Remark. When these are minimal, Picard numberρ(S ) = 2. Question 1. Is there a DP1 withρ = 1 that is unirational?
Conjecture(Batyrev–Manin–Peyre). Suppose thatS is a Del Pezzo surface of degree1over a number field k with a rational point and Picard numberρ. Then there is a nonzero constantc such that for every small enough nonempty open subsetU ⊂ S we have
#{ P ∈ U(k) : H−K(P) ≤ B } ∼ cB(log B)ρ−1
as B → ∞.
Nonzeroness comes from Peyre’s description ofc or Colliot-Th´el`ene’s conjecture that Brauer–Manin is the only obstruction to weak approximation on rational varieties.
Every Del Pezzo surfaceS /k of degreed = 1 is isomorphic to a smooth sextic inP(2, 3, 1, 1), with coordinates x , y , z, w, given by
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6
withai ∈ k[z, w ]homogeneous of degree i. (And vice versa.)
Linear system| − KS|induces rational mapS 99K P1(z, w ). Unique base pointO = [1 : 1 : 0 : 0].
Curves in| − KS|are fibers of BlO(S ) → P1 (anticanonical fiber).
Theorem(V´arilly-Alvarado). LetA, B be nonzero integers, and letS be the Del Pezzo surface of degree 1overQ given by
y2 = x3+ Az6+ Bw6.
Assume that Tate–Shafarevich groups of elliptic curves overQwith j-invariant 0 are finite. If 3AB is not a square, or ifA andB are relatively prime and9 - AB, then S (Q)is Zariski dense inS. Question. What about y2 = x3+ 243z6+ 16w6 ?
Theorem(Salgado–vL). Let S ⊂ P(2, 3, 1, 1)be a Del Pezzo surface of degree1over a:::::::number:::::field k. Suppose F is a smooth::::::: anticanonical fiber with a pointP ∈ F (k) of order > 2 that does not lie on six exceptional curves. SetU = P(2, 3, 1, 1) \ Z (z, w ) and letC denote the curve of those sections of the projection U → P1(z, w )that meetS at the pointP with multiplicity at least5. If #C(k) = ∞, thenS (k)is Zariski dense.
Remark. If the order of P is 3, then #C(k) = ∞ for free. For V´arilly-Alvarado’s example
y2 = x3+ 243z6+ 16w6
there is a3-torsion point[0 : 4 : 0 : 1], but it lies on nine exceptional curves...
The coefficients of the curveC associated to the point [−63 : 14 : 1 : 5]are too large to find points...
Elkies’ proof for V´arilly-Alvarado’s example. Take affine part y2 = x3+ 243 + 16t6,
and sety = v + 4t3 to obtain
v2+ 8t3v = x3+ 243.
Now projection ontov-line gives fibration into cubics. The point (t, x , y ) = (5, −63, 14)gives a cubic with infinitely many rational points, most of which have infinite order on their anticanonical fiber. Done!
Elkies: When I tried to generalise this construction, it turned out I needed a3-torsion point. Maybe it can be done for all rational torsion points (on their anticanonical fiber).
Theorem(Bulthuis-vL). Let S be a Del Pezzo surface of degree 1 over a number fieldk. Suppose F is a smooth anticanonical fiber with a pointP ∈ F (k)of finite order n > 1. Then the linear system
| − nKS − nP| = { D ∈ | − nKS| : µP(D) ≥ n } induces a fibrationϕ : BlP(S ) → P1 of curves of genus1. If
1) some irreducible fiber ofϕhas∞ manyk-rational points, or 2) there is aQ ∈ S (k) \ {P}such that the fiber GQ = ϕ−1(ϕ(Q))
is smooth, and on the elliptic curve(GQ, Q), the sum of the
points aboveP has infinite order, thenS (k)is Zariski dense.
Moreover, there is a nonempty Zariski open subsetU ⊂ S such that everyQ ∈ U(k)satisfies the conditions of 2).
Consequence. Suppose S has a point that has finite order on its anticanonical fiber. IfS (k)is Zariski dense inS, then there is an easy proof that this is the case.
Setπ : S0= BlP(S ) → S, except. curveE, strict transf. F0 of F.
1. | − nKS− nP| ↔ | − nπ∗KS − nE | = |nπ∗F − nE | = |nF0|.
2. dim |mF0| = 0for0 ≤ m < n, anddim |nF0| = 1. 3. Base locus of |nF0|is empty: 1-dim’l components are F0,
contradicting 2. No base points asF02= (π∗F − E )2 = 0. 4. Bertini Theorem: almost all curves in|nF0|smooth.
5. All curves connected: Stein factorisation gives S0 → P1 with
fibers H with aH ∼ nF0 for somea, contradicting 2. 6. Almost all curves geometrically integral.
7. Genus of smooth D ∈ |nF0|is 1: we have −KS0 ∼ F0, so
2g (D) − 2 = D(D + KS0) = D(D − F0) = 0.
8. Fibers of ϕare not in torsion locus of anticanonical fibration: torsion-locus does not self intersect in smooth fibers.
9. Equivalent condition on Q: the divisor n(Q) −P
To finddim |mF0| = dim H0(S0, O
S0(mF0)) − 1, we consider the
embeddingι : F0 ,→ S0 and exact sequence (idea Adam Logan)
0 → OS0(−F0) → OS0 → ι∗OF0 → 0
of sheaves onS0. Twisting by OS0(mF0)gives
0 → OS0((m − 1)F0) → OS0(mF0) → ι∗OF0 ⊗ OS0(mF0) → 0 .
LetF∞ be an other anticanonical fiber and F∞0 = π∗F∞ its strict
transform. ThenF0 ∼ F0
∞− E, so
ι∗OF0⊗OS0(mF0) ∼= ι∗OF0⊗OS0(mF∞0 −mE ) ∼= ι∗OF0(mO −mP).
We obtain
0 → OS0((m − 1)F0) → OS0(mF0) → ι∗OF0(mO − mP) → 0 .
0 → H0S0, O((m − 1)F0) → H0 S0, O(mF0) → H0(F0, mO − mP) → H1S0, O((m − 1)F0) → H1 S0, O(mF0) → H1(F0, mO − mP) Using (for i ∈ {0, 1}) dim Hi(F0, mO − mP) = ( 1 if n|m 0 otherwise and dim Hi(S0, OS0) = ( 1 if i = 0 0 if i = 1 ,
we find, by induction, that (for i ∈ {0, 1} and0 < m < n)
dim Hi(S0, OS0(mF0)) =
(
1 if i = 0 0 if i = 1 . Finally, for m = n, we obtain
