Shuijing Crystal Li Rice University Mathematics Department 1 Rational Points on del Pezzo Surfaces of degree 1 and 2

2 Rational Points on algebraic surface Chevalley-Warning Theorem Example What is del Pezzo surface? Definition Geometric structure of del Pezzo surface Rational Points on del Pezzo surface Rational points on cubic surface Rational Points on del Pezzo surface of degree 1 and 2

3 Chevalley-Warning Theorem Question: When it is sharp? When do we have unique solution?

4 Pierre de Fermat Fermat’s Cubic Surface

Examples of Cubic Surfaces The Clebsch Diagonal Surface The Clebsch Diagonal Surface is one of the most famous cubic surfaces because of its symmetry and the fact that it's the only one with ten Eckardt Points.Eckardt Points Defining equation 0= x 3 + y 3 + z 3 + w 3 - (x+y+z+w) 3 From the picture below the surface, we can see that there are ten Eckardt Points (points, where three lines meet in a point).Eckardt Points The Clebsch Diagonal Surface The Clebsch Diagonal Surface is one of the most famous cubic surfaces because of its symmetry and the fact that it's the only one with ten Eckardt Points.Eckardt Points Defining equation 0= x 3 + y 3 + z 3 + w 3 - (x+y+z+w) 3 From the picture below the surface, we can see that there are ten Eckardt Points (points, where three lines meet in a point).Eckardt Points 10 Eckardt points and 27 lines 5

6 Theorem ( Iskovskikh) Now, let’s try to find some examples with unique rational point. First, let’s look at some hypersurfaces of dimension 2: del Pezzo surfaces del Pezzo surfaces Given a rational variety X of dimension 2 over perfect field k, at least one of the following happens: a)X is birational to a conic bundle over a conic b)X is k-birational to a del Pezzo surface. Hence, we understand the del Pezzo surfaces, we understand almost all the rational surfaces. How wonderful is that? Hence, we understand the del Pezzo surfaces, we understand almost all the rational surfaces. How wonderful is that? Classic Definition We can think of the del Pezzo surfaces as subset of projective space given as the zero locus of some homogeneous polynomials. Algebraic Point of View:

7 Another way to study the geometry of an algebraic surface is to look at the curves that lie on the surface. “Lines” means the curves of minimal degree, and will be replaced by exceptional curves or (- 1)-curves, since any line satisfies (L,L)= -1 Many interesting arithmetic questions are connected with the class of del Pezzo surfaces, as such surfaces are geometrically rational (i.e. rational over the algebraic closure). It is especially interesting to look at problems concerning the question about the existence of k-rational points, where k is a non-closed field. Theorem (Yu.I.Manin,1986) The defining equation of del Pezzo surface

8 Amazingly, the answer is YES, and there are finitely many of them! How many? How are they configured? Question: Are there any “lines” on del Pezzo surfaces? Theorem Every del Pezzo surface has only finitely many exceptional curves, and their structure is independent of the location of the points blown up, provided that they are general.

9 Let’s compute the lines on Fermat cubic surface. Now if we set

10 Q : Does there exist a cubic surface with a unique k-rational point?

11 Del Pezzo Surface of Degree 1 Q : Does there really exist a del Pezzo surface of degree 1 with unique k-rational point over some local field? Algebraic Way: Running through all possible coefficients to find unique solution Geometric Way: Using Weil’s Theorem and the trace information

12 Theorem ( A. Weil)

13 Possible Unique rational point over F3 Possible Unique rational point over F7 Possible Unique rational point over F4 Possible Unique rational point over F2

14 Using computer program running through all possible coefficients, then find all the corresponding solutions to each equation projectively. Let X be smooth del Pezzo surfaces of degree 1 defined as above, then X has at least 3 rational points over any finite field Theorem

15 Del Pezzo Surface of degree 2 Same Question Same Approach Different Answer

16 Finally, using computer program running through all possible coefficients.

17 Theorem

18 Remark We can easily seen from the analysis above, the structure of del pezzo surfaces of degree d>=4 are relatively easy. For 7>=d>=5, all Del Pezzo surfaces of the same degree are isomorphic.