Weak Approximation for General Degree Two Del Pezzo Surfaces
We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two singularities in order to prove weak approximation for degree two del Pezzo surfaces with square-free discriminant.
A standard question in Arithmetic Geometry is, “Does a variety defined over a number field contain -rational points, and if so, how are these points distributed on ?” We say that satisfies weak approximation if for any finite set of places of and points of over the completion of at these places, there exists a -rational point of which is arbitrarily close to these points. In this paper, we study varieties defined over the function field of a smooth curve instead of a number field. In this context , rational points correspond to sections of fibrations over a curve, and proving weak approximation corresponds to finding sections with prescribed jet data in a finite number of fibers.
The existence of sections of rationally connected fibrations was proven by Graber, Harris and Starr . Kollár, Miyaoka, and Mori proved the existence of sections through a finite set of prescribed points in smooth fibers  IV.6.10 and  2.13. The existence of sections with prescribed finite jet data through smooth fibers, i.e. weak approximation at places of good reduction, was proven by Hassett and Tschinkel . In the same paper Hassett and Tschinkel conjectured that a smooth rationally connected variety defined over the function field of a smooth curve satisfies weak approximation even at places of bad reduction.
There are a few cases where weak approximation over the function field of a curve is known to hold at all places. Colliot-Thélène and Gille proved weak approximation holds for stably rational varieties, connected linear algebraic groups and homogeneous spaces for these groups,homogeneous space fibrations over varieties that satisfy weak approximation . In particular they prove that conic bundles over and del Pezzo surfaces of degree at least four satisfy weak approximation at all places. The cases of del Pezzo surfaces of degree less that four are still open. It is known that cubic surfaces with square-free discriminant satisfy weak approximation even at places of bad reduction  . This paper address weak approximation at places of bad reduction for degree 2 del Pezzo surfaces.
Let be an algebraically closed field of characteristic zero, a smooth curve over with function field , and the smooth projective model of with .
Let be a smooth degree two del Pezzo surface over , and a proper model of (i.e. an algebraic space over flat over with generic fiber .) Suppose the singular fibers of are degree two log del Pezzo surfaces with at most two singularities. Let be the locus where is smooth. If there exists a section , then the sections of satisfy approximation away from .
The proof of Theorem 1 relies on approximation results of  and extending the rational connectivity results of  to show that in is a degree two log del Pezzo surface with one singularity, then each smooth point of is contained in a free proper rational curve contained in the smooth locus of . This proof also shows that the sections of satisfy approximation away from when the singular fibers are degree two del Pezzo surfaces with the following singularity types:
The rationality of and properness of imply the existences of sections of . When the model is regular, all sections of are contained in the smooth locus, so we find:
Let be a smooth degree two del Pezzo surface over . If admits a regular proper model whose singular fibers are degree two log del Pezzo surfaces with two singularities, then weak approximation holds for away from .
This corollary is applicable whenever is a smooth projective curve because will always admit a regular proper model. There exist smooth degree two del Pezzo surfaces that admit models with worse than singularities. For example the family
over the -line has a fiber with worse than even rational double points when . But, Corollary 2 says weak approximation holds for ‘generic’ degree two del Pezzo surfaces.
Let be a normal projective surface of degree 2 over with ample anti-canonical sheaf . Furthermore, suppose that the singularities of are at most rational double points. We call a degree 2 log del Pezzo surface with rational double points. Such an is a double cover of branched over a quartic curve without multiple components thus may be thought of as a quartic hypersurface in the weighted projective space defined by the equation where  .
Assume that is a smooth degree two del Pezzo surface defined over a smooth curve , and is a model of with smooth total space. We would like to define the discriminant for this model, so we replace the singular space with its minimal resolution . Now the fibers of are degree four hypersurfaces in . Suppose is a degree four hypersurface in and contains . Let be the proper transform of in . Then will intersect the exceptional along a conic of self-intersection .
Let be the differential map from a rank three vector bundle to a rank one vector bundle over . Let be the degeneracy locus of in and define the discriminant of to be the image of in B. The expected dimension of is zero, but the cone point can lead to components of dimension one as seen in the previous paragraph. The excess intersection formula  6.3 tells us that the one dimensional components of have multiplicity two in . We also know that in the multiplicity of at each zero dimensional component of is the sum of the Milnor numbers of the singularities in the corresponding fiber  7.1.14. We say that the discriminant of is square-free if each component has multiplicity one, i.e. the fibers of each have at most one singularity.
Suppose is a smooth degree two del Pezzo surface defined over a smooth curve . If the discriminant of is square-free, then sections of satisfy approximation away from .
Acknowledgments: I am grateful to my thesis advisor Brendan Hassett for the many conversations we have had about this topic. I also benefitted from talking to Damiano Testa and Brad Duesler. This material is based upon work supported by the National Science Foundation under Grants 0134259 and 0240058.
2. Weak Approximation
Let be a number field or the function field of a curve . Let be a finite set of places of that contains the archimedean places . For each place of , let denote the -adic completion of . Let be an algebraic variety defined over and the set of -rational points. One says that weak approximation holds for away from if for any finite set of places of , for , and -adic open subsets , there is a rational point such that its image in each is contained in .
By restricting ourselves to the case of function fields, we can formulate a more geometric description of weak approximation. Let be a smooth curve over an algebraically closed field of characteristic zero with function field . Let be a smooth projective model of and set . Let be a smooth proper variety over , a proper flat model (existence was proven in ), and the smooth locus of . Since is a proper morphism, sections of correspond to -rational points of . The analogue of local points are the -jets defined below.
Let be a proper model of over :
An admissible section of is a section .
An admissible -jet of at is a section of
whose image is a smooth point of .
An approximable -jet of at is a section of
that may be lifted to a section of where Spec and .
Note that Hensel’s Lemma implies that every admissible -jet is approximable, and every section is admissible when the model is regular. We can now formulate a geometric notion of weak approximation:
We say that satisfies weak approximation away from if any finite collection of approximable jets of can be realized by a section . If is a regular model this is equivalent to the condition that any
collection of admissible jets of can be realized by a section
When we want to refer to a specific model , we say that sections of satisfy approximation away from if any finite collection of approximable jets of can be realized by a section .
3. Notions of Rational Connectivity
A variety is rationally connected if there is a family of proper algebraic curves whose fibers are irreducible rational curves and a cycle morphism such that
When is defined over an uncountable algebraically closed field , rational connectivity is equivalent to the condition that any two very general points in can be joined by an irreducible projective rational curve contained in  IV.3.6.
Let be a smooth algebraic variety and a nonconstant morphism, so we have an isomorphism
for suitable integers Then is free (resp. very free) if each (resp. ).
When is a smooth variety over an algebraically closed field, being rationally connected is equivalent to containing one very free curve  IV 3.7. Suppose we are in this situation. Then there exists a unique largest nonempty subset such that for any finite collection of distinct points in there is a very free rational curve contained in which contains these points as smooth points. Moreover, any rational curve that meets is contained in  IV.3.9.4. There are no known examples where . The case of leads to the following definition.
Definition 8 ( 14).
A smooth rationally connected variety is strongly rationally connected if any of the following conditions hold:
for each point , there exists a rational curve joining and a generic point in ;
for each point , there exists a very free rational curve containing .
for any finite collection of points , there exists a very free rational curve containing the as smooth points;
for any finite collection of jets
supported at distinct points , there exists a very free rational curve smooth at and containing the prescribed jets.
We note here that in Definition 14 of  property reads ‘free’ and not ‘very free,’ but having a free curve through every point has not been proven to be equivalent to any of the other conditions. When the variety is a surface, the word ‘very free’ can be replaced by ‘free’ in property .
A smooth rationally connected surface is strongly rationally connected if and only if for each point , there exists a free rational curve containing .
When there is a family of rational curves through dominating , one of these curves is guaranteed to be free [10, 1.1]. Now suppose there is a free curve through every point in . For any , let be a free rational curve containing . Let be a free curve containing the generic point. Since and are free, we know that . By the Hodge Index Theorem . Thus the curves and intersect in . We can smooth this curve to a rational curve joining and the generic point [9, II.7.6]. ∎
The main point of this paper is to prove that the smooth locus of a degree two log del Pezzo surface with an singularity is strongly rationally connected because Hassett and Tschinkel proved the following theorem:
( 15) Let be a smooth morphism whose fibers are strongly rationally connected. Assume that has a section. Then sections of satisfy approximation away from .
4. Combs and Free Curves
Our main goal of this paper is to find a free rational curve through any smooth point in . Suppose we have a surface containing many very free rational curves. Suppose also that we have found a rational curve through a point which is not free. This section shows how we use combs to create a free rational curve through the point .
A comb with teeth is a connected and reduced curve with irreducible components such that:
is smooth and called the handle ;
every is isomorphic to and these are calles the teeth;
the only singularities of are ordinary nodes;
every intersects in a single point, and when .
A rational comb is a comb whose handle is a smooth rational curve.
We construct a comb whose handle is an arbitrary nonfree rational curve on a smooth quasi-projective variety . Assume there exists several very free rational curves meeting at distinct smooth points . We want to deform the union into a very free rational curve. One construction of such a comb was considered by Kollár, Miyaoka, and Mori, see [9, II.7.9, II.7.10]. A more recent construction of a comb that deforms nicely comes from the techniques of Graber, Harris, and Starr. This construction considers deformations in the Hilbert scheme of . We include here some essential facts about Hilbert schemes which are used in constructing the comb that frees .
There is a -scheme Hilb parameterizing closed subschemes of . Assume is smooth, and let be a closed subscheme of given by the ideal sheaf . Assume is a local complete intersection, i.e. is locally generated by codim elements. Then is locally free on and its dual is the normal bundle of in . In this situation the following hold [9, I.2.8]:
If H, then Hilb is smooth at .
Suppose there are disjoint smooth rational curves each meeting a rational curve transversely in a single point , . Fix a point that is not one of the nodes . Then we can deform their union into an irreducible curve provided the following conditions hold [4, 2.6]:
The sheaf is generated by global sections.
We can see this by considering the sequence
where is a torsion sheaf supported at the points . If is generated by global sections, then we can find a global section H such that, for each , the restriction of to a neighborhood of is not in the image of . This means corresponds to a first-order deformation of that smoothes the nodes of . If the second condition holds, there is no obstruction to lifting our first-order deformation to a global deformation of smoothing the nodes.
For a general fixed point , we would like to create a comb which deforms into an irreducible free curve while still passing through . In order to achieve this, we replace the conditions above with new conditions:
The sheaf is generated by global sections.
( 27) Let be a smooth projective variety of dimension at least 3 over an algebraically closed field. Let be a smooth irreducible curve and a line bundle on . Let be a very free rational curve intersecting and let be a family of nodal rational curves on parameterized by a neighborhood of in Mor. Then there are curves such that is an immersed comb and satisfies the following conditions:
The sheaf is generated by global sections.
H, where is the unique line bundle on that extends and has degree 0 on the .
Thus on a variety of dimension at least three containing many very free curves, rational curves meeting the very free curves can be freed. We can generalize this result a little.
Let be a smooth surface over an algebraically closed field. Let be an irreducible rational curve and a point on . Let be a very free rational curve intersecting and let be a family of rational curves on parameterized by a neighborhood of Mor. Then there are curves such that is a comb whose nodes can be smoothed to create a free rational curve containing .
We modify the smoothness of and dimension of in Theorem 12.
Step 1. Reduction to the case where is a smooth rational curve.
Suppose is singular. Then we will work with the normalization . When considering the Hilbert Scheme, it is essential that be embedded in , so we take an embedding and consider the diagonal map This is an embedding with image isomorphic to . Let denote the projection from to the first factor. Then . Any moving of can be projected to to give a deformation of . From now on assume is a smooth, irreducible, rational curve.
Step 2. Reduction from surfaces to three-folds.
Suppose is a smooth surface and is a morphism. Consider the three-fold and the map . If we let be the projection onto the second factor, then Since is smooth, the map is surjective . Thus if is ample, then is ample.
Now once we find a rational curve in containing a given smooth point , we know how to free it. So what is left to do is to find rational curves through points.
5. Proof of the Main Theorem
Keel and McKernan proved the smooth locus of any log del Pezzo surface is rationally connected . As stated in the Section 3, the main contribution made by this paper is the following refinement for degree two del Pezzos:
Let be a degree two log del Pezzo surface with two singularities and the smooth locus of . Then is strongly rationally connected.
Let be a double cover of branched over a quartic plane curve . Since has two singularities, has two nodes  III.7.1. In order to prove the theorem, for every point we will find a free curve containing .
Consider arbitrary . Let . We must find a proper curve in which lifts to a free rational curve in . The simplest plane curves are lines, so we can try to find lines through that lift to rational curves. By the Hurwitz formula we see that a generic line in will lift to a curve of genus one in . However, lines simply tangent to will lift to rational curves. Thus, we consider lines tangent to the branch curve .
First we consider the case when is not contained in . Let such that . Let be the normalization of and consider the projection from . Then restricted to the the genus two curve is a four-to-one cover of the projective line with ramification index equal to ten. Since the ramification number is ten, there are at least four lines through that are tangent to and three of these lines will not contain the singular point . We only need to pick one of these lines and analyze the four ways a line can be tangent to a quartic: one simple tangent; one three tangent; one four-tangent; or two simple tangents.
Next suppose that is a smooth point on the quartic . If the tangent line of at does not contain the singular point, , then we can use the tangent line at to find a free curve through with the analysis below. The tangent line at passing through through is not actually a problem. If we consider projection from again, we have a three to one cover of the line with ramification index equal to eight and are guaranteed that is contained in at least four lines tangent to . We simply pick one of the lines missing the singular point.
Now we analyze the four cases of tangency between a line an a quartic cure, and construct free curves in from lifts of the lines.
One Simple Tangent.
Suppose that we can find a line through that is
simply tangent to at a smooth point of and intersects the
quartic in two other distinct points. By the Hurwitz formula, we
know that the normalization, , of the curve lying over is rational, thus is rational. Let
be the image of the projective line under the map .
We have an exact sequence
We know that , and by the double point formula deg. The extension
is given by an element of Ext H. Thus, the sequence splits and , and is a free curve through the point .
We can even create combs with as the handle in order to find very free curves through . Since is rationally connected, we know that contains a very free rational curve . The Hodge Index Theorem and the the fact the and are positive imply that and intersect. By Proposition 10, we see that there exists a very free curve through .
One Inflectional 3-Tangent. Now assume we can find a line through that is three-tangent to at a smooth point. As before, we let be the rational curve lying over and containing . With the same set up as the previous case, we return to the exact sequence,
Again we find that , and we have a free rational curve
through the point .
One Inflectional 4-Tangent. Suppose we have a line through intersecting the quartic at only one smooth point. Then is four tangent to at the point of intersection, and the corresponding curve above will have a tacnode above the intersection. Because the arithmetic genus of the normalization of the curve is -1, we know must be the union of two irreducible rational curves . Instead of considering the entire curve , we shall focus on the component passing through the original point .
Because each is mapped one-to-one onto the line in by the anti-canonical sheaf of , we know . Adjunction then implies that the self intersection, hence the degree of the normal sheaf, of each is -1. Hence, our curve is not free.
We use the dual curve of to show that we can use a comb as in Proposition 12 to free . For each point
let be the tangent line to at . Consider
as a point in the dual projective plane . There is a
map from to its dual curve which
sends a point to . The singularities of are:
nodes corresponding to bitangents to , cusps corresponding to
inflectional 3-tangents of , tacnodes corresponding to
inflectional 4-tangents of . The dual curve to a quartic with one node is a
planar curve of degree 10 and has only a finite number of
singularities. Thus, for an infinite number of points on the line
, there are lines through these points that are simply tangent to
and intersect transversely. These lines lift to free curves
in as shown above, and can be made into the very free curves that we need in order to apply the proposition.
Two Simple Tangents. Finally suppose there exists a line through that is tangent to at two distinct nonsingular points. Then the curve lying above will have a node above each of these tangencies. The arithmetic genus of the normalization of is -1, so we know is the union of two rational curves . As in the case above, we find that the degree of the normal sheaf of each is -1. Suppose is the component that contains the point . We can use very free curves intersecting to create a comb to free . The resulting curve will be a free rational curve through as desired. ∎
As stated in the introduction, this proof can be easily modified to show that the smooth locus of a degree two del Pezzo surface is strongly rationally connected when the singularity type of the surface is one of the following:
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