Dolgachev Research Articles

A FEW EXPLICIT EXAMPLES OF COMPLEX DYNAMICS OF INERTIA GROUPS ON SURFACES—A QUESTION OF PROFESSOR IGOR DOLGACHEV

We give a few explicit examples which answer a question of Professor Igor Dolgachev on complex dynamics of the inertia group of a smooth rational curve on a projective K3 surface and its variants for a rational surface and a non-projective K3 surface.

Computational algebraic geometry and its applications

This issue of the Applicable Algebra in Engineering, Communication and Computing presents selected papers on work presented at, or related to, the first two meetings of the Michigan Computational Algebraic Geometry group, namely MCAG 2012, MCAG 2013. MCAG conferences are a yearly event held in one of the public universities in Michigan. Their aim is to bring together algebraic geometers from academia and industry and to foster further cooperation. MCAG-group is a network of algebraic geometers in Michigan and surrounding areas in order to encourage collaboration, exploration of new applications, and stronger connections to industry. The group organizes yearly conferences in different Michigan institutions, training sessions for students, and encourages collaboration among different mathematics departments from Michigan. During the first two MCAG conferences, there were hundred of participants from all public research universities in Michigan. There were also participants from Ontario, Canada and many Midwest sates. Among invited speakers were Igor Dolgachev (University of Michigan), William Fulton (University of Michigan), Gene Freudenburg (Western Michigan University), Rajesh Kulkarni (Michigan State University), Harm Derksen (University of Michigan), Charles Wampler (General Motors), Juan Migliore (University of Notre Dame), Annegret Paul (Western Michigan University), Gregory Pearlstein (Michigan State University), Tony Shaska (Oakland University), David Speyer (University of Michigan), Jan Verschelde (University of Illinois—Chicago), Fabrizio Zanello (Michigan Tech.), Dan Erman (University of Michigan), Cynthia Vinzant (University of Michi-

Del Pezzo surfaces of degree 4 and their relation to Kummer surfaces

In this note, which has little pretence to originality, we clarify the relation between the geometry of del Pezzo surfaces of degree 4 and their realization as the zero set of two quadratic forms in five variables. We also review the classical description of the desingularized Kummer surface K constructed from the Jacobian J of a curve C of genus 2 as the zero set of three quadratic forms in six variables (Plucker, Kummer, Klein [7], [6], see [5] or [3] for a modern treatment). If C has a rational Weierstrass point, a partial diagonalization of this system gives rise to a natural projection onto a hyperplane, defining a finite morphism π : K → X of degree 2 onto a del Pezzo surface X of degree 4 (see [9, §6]). We show that X is the blow-up of Pk in the images of the five other Weierstrass points of C under the embedding of Pk as a conic in Pk. The morphism π sends the 16 lines on K to the 16 lines on X, and is equivariant with respect to the action of the subgroup of 2-division points J [2] ⊂ J . Thus π gives rise to a morphism from the twisted Kummer surface to the twisted del Pezzo surface. In our presentation it is obvious that all del Pezzo surfaces of degree 4 can be obtained in this way, an observation made by Victor Flynn in [4]. The fact that any 2-covering of J maps to a del Pezzo surface of degree 4 was first observed in [2], and used in [2], [1] and [9] to construct and visualize elements of order 2 in the Tate–Shafarevich group of J over Q using the theory of the Brauer– Manin obstruction on del Pezzo surfaces of degree 4. It was the author’s desire to understand the geometry behind these calculations that prompted him to write this note. I would like to thank Igor Dolgachev for useful discussions.

Correspondence between K3 surfaces ( With an appendix by Igor Dolgachev)

In this paper we study the existence of correspondences between K3 surfaces X(k, m, n) with k, m, n ∈ N, Picard number 17, and transcendental lattices T(k, m, n) ∼= U(k)⊕U(m)⊕〈−2n〉. In a fundamental paper, Mukai [Mu] showed that correspondences between K3 surfaces exist if the transcendental lattices are Hodge isometric over Q. This construction holds if the Picard number of the surfaces is greater than or equal to 11. Nikulin [N2] later improved this result, obtaining the lower bound 5 for the Picard number. The aim of our work is to realize examples of K3 surfaces with transcendental lattice that are not Hodge isometric but such that a correspondence between them already exists. This in particular implies the existence of an algebraic cycle on the middle cohomology of the product of two surfaces arbitrarily chosen in the constructed family. In Sections 2 and 3 we recall some basic notions and results on lattices and correspondences. In Section 4 we consider a generic genus-2 curve and we show the existence of a correspondence between the Jacobian of the curve and a K3 surface with isomorphic transcendental lattice. Since this construction involves a second K3 surface whose transcendental lattice has quadratic form multiplied by 2, in Sections 5 and 6 we generalize this first example. First, we construct K3 surfaces “twisting” each direct summand of the transcendental lattice of the Jacobian by natural numbers. Then we find correspondences between them using both Mukai’s theorem and Shioda–Inose structures that translate the problem into a problem of looking for isogenies between abelian varieties. In this way we prove in Theorem 6.3 that all the K3 surfaces X(k, m, n) are in correspondence to each other. Finally, in Theorem 6.5 we show the existence of a correspondence between a general K3 surface of Picard rank 17 and a Kummer surface of the same rank having transcendental lattices that are Q-Hodge isomorphic. In the Appendix, Igor Dolgachev realizes a geometric correspondence between the K3 surfaces of Section 4.