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

2 What is del Pezzo surface? Definition Geometric structure of del Pezzo surface Examples of del Pezzo surface How could we see/understand it? An overview of Cubic Surfaces Rational Points on del Pezzo surface Rational points on cubic surface Rational Points on del Pezzo surface of degree 1 and 2

3 An open problem zy x

4 Geometry Variety Algebra Solving equations

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6 Theorem ( Iskovskikh) Q: Among numerous algebraic varieties, why do we care about “ 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?

Our special algebraic variety : del Pezzo surfaces Our special algebraic variety : del Pezzo surfaces 7 Definition From now on, we can then think of the del Pezzo surfaces as subsets of projective space given as the zero locus of some homogeneous polynomials. Algebraic Point of View: Remark

8 Let’s jump to geometry Many interesting arithmetic questions are connected with the class of del Pezzo surfaces, as such surfaces are geometrically rational (i.e. rational over the complex field). 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.

After learning about the blow-up and using it to construct del Pezzo surfaces, we will turn to study their geometric structure. 9 Let X be a del Pezzo surface of degree d. (a) 1 ≤ d ≤ 9. (b) (Classification) If the base field is separably closed, either:  X is isomorphic to the blow-up of a projective plane at k = 9 − d points, or  X is isomorphic to P 1 xP 1 (and d = 8) In particular, if d = 9, then X is isomorphic to the projective plane P 2 (the blow-up of P 2 at no points). (c) (Converse) If X is the blow-up of a projective plane at k = 9 − d points in generic position (no three points collinear, no six on a conic, no eight of them on a cubic having a node at one of them), then X is a del Pezzo surface. Theorem(Yu.I.Manin) Classification of del Pezzo surfaces Another way to study the geometry of an algebraic surface is to look at the curves that lie on the surface. Since our del Pezzo surfaces can be embedded into projective space, we can study whether any “lines” in projective space are completely contained in our surface. Question: Are there any “lines” on del Pezzo surfaces?

10 Amazingly, the answer is YES, and there are finitely many of them! How many? How are they configured? From now on, “lines” means projective lines, and will be replaced by exceptional curves or (-1)-curves. 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.

11 Remark We can easily seen from the table 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. Back to Algebra and answer a previous question: Theorem (Yu.I.Manin,1986) The defining equation of del Pezzo surface

An overview of Cubic Surfaces Cubic Surfaces A cubic surface is the vanishing set of a homogenous polynomial of degree 3 in P 3, i.e. it consists of all (x:y:z:w) in P 3 with: a 0 x 3 + a 1 x 2 y + a 2 x 2 z +... a 18 z 2 w + a 19 w 3 = 0History In the 19 th century, mathematicians started to study the structure of such vanishing sets of polynomials of different degrees in P 3, called algebraic surfaces. It turned out, that each generic cubic surface contains 27 straight lines. From this starting point, a lot of mathematicians have studied cubic surfaces and the structure of the 27 lines upon it. In 1861, Clebsch showed, that the defining equation of a cubic surface can be put, in a unique way, in the so called pentahedral form, which allows us to calculate the equations of the 27 lines directly from the equation. 12

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 13

Examples of Cubic Surfaces The Cayley Cubic The Cayley Cubic Surface contains four double points (which is the maximum number for any cubic surface).double points Defining Equation is 4( x 3 + y 3 + z 3 + w 3 ) - (x+y+z+w) 3. From the picture below the surface, one sees immediately, that there are four double points on the surface (each one corresponds to a set of three points on a line in the plane). double points The Cayley Cubic The Cayley Cubic Surface contains four double points (which is the maximum number for any cubic surface).double points Defining Equation is 4( x 3 + y 3 + z 3 + w 3 ) - (x+y+z+w) 3. From the picture below the surface, one sees immediately, that there are four double points on the surface (each one corresponds to a set of three points on a line in the plane). double points 4 double points 14

15 Theorem ( Jano’s Kollar, 2000) Rational Points on Cubic Surface Rational Points on Cubic Surface on Cubic Surface Theorem ( Unirationality of del Pezzo surface)

16 Q : Does there exist a cubic hypersurface with unique k-rational point? From Kollar’s theorem, we discover that the unirationality of an algebraic surface is closely related to the existence of k-rational point.

17 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? 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

18 Theorem ( A. Weil)

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

20 Before running to computer for help, is there any other way? YES! Geometry

21 Finally, using computer program running through all possible coefficients, then I found: there is no degree 1 del Pezzo surface with unique rational point over any local field

22 Theorem 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

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

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

25 Theorem

26 Acknowledgement Professor Brendan Hassett Professor Robert Hardt Professor Ron Goldman Funding: Supported by NSF