Kummer surface

In algebraic geometry, Kummer's quartic surface is an irreducible algebraic surface over a field $K$ of characteristic different then 2, which is a hypersurface of degree 4 in $\mathbb {P} ^{3}$ with 16 singularities; the maximal possible number of singularities of a quartic surface. It is a remarkable fact that any such surface is the Kummer variety of the Jacobian of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution $x\mapsto -x$ . The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface.

Geometry of the Kummer surface

Singular quartic surfaces and the double plane model

Let $K\subset \mathbb {P} ^{3}$ be a quartic surface, and let $p$ be a singular point of this surface. Identifying the lines in $\mathbb {P} ^{3}$ thorugh the point $p$ with $\mathbb {P} ^{2}$ , we get a double cover from the blow up of $K$ at $p$ to $\mathbb {P} ^{2}$ ; this double cover is given by sending $q\neq p\mapsto {\overline {pq}}$ , and any line in the tangent cone of $p$ in $K$ to itself. The ramification locus of the double cover is a plane curve $C$ of degree 6, and all the nodes of $K$ which are not $p$ map to nodes of $C$ .

By the genus degree formula, the maximal number possible number of nodes on a sextic curve is obtained when the curve is a a union of $6$ lines, in which case we have 15 nodes. Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that $p$ is simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.

Since $p$ is a simple node, the tangent cone to this point is mapped to a conic under the double cover. This conic is in fact tangent to the six lines (w.o proof). Conversely, given a configuration of a conic and six lines which tangent to it in the plane, we may define the double cover of the plane ramified over the union of these 6 lines. This double cover may be mapped to <mathbb>\mathbb{P}^3</math>, under a map which blows down the doubel cover of the special conic, and is an isomorphism elsewhere (w.o. proof).

The double plane and Kummer varieties of Jacobians

Starting from a smooth curve $C$ of genus 2, we may identify the Jacobian $Jac(C)$ with $Pic^{2}(C)$ under the map $x\mapsto x+K_{C}$ . We now observe two facts: Since $C$ is a hyperelliptic curve the map from the symmetric product $Sym^{2}C$ to $Pic^{2}C$ , defined by $\{p,q\}\mapsto p+q$ , is the blow up along the canonical divisor. Moreover, the canonical map $C\to |K_{C}|^{*}$ is a double cover. Hence we get a double cover $Kum(C)\to Sym^{2}|K_{C}|^{*}$ . This double cover is the double cover which appeared above, where the 6 lines are the images of the odd symmetric theta divisors on $Jac(C)$ , while the conic is the image of the blown-up 0. Both the conic, and each of the six lines is an image of the canonical system of $C$ . There is a 1-1 correspondence between pairs of odd symmetric theta divisors and 2-torsion points on the Jacobian given by the fact that $(\Theta +w_{1})\cap (\Theta +w_{2})=\{w_{1}-w_{2},0\}$ , where $w_{1},w_{2}$ are Weierstrass points (which are the odd theta characteritics in this cae). Hence the branch points of the canonical map $C\mapsto |K_{C}|^{*}$ appear on each of these copies of the canonical system as the intersection points of the lines and the tangency points of the lines and the conic.