A ribbon is a first-order thickening of a non-singular curve. Motivated by a question of Eisenbud and Green, we show that a compactification of the moduli space of line bundles on a ribbon is given by the moduli space of semi-stable sheaves. We then describe the geometry of this space, determining the irreducible components, the connected components, and the smooth locus.
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field
$k$
, generalizing the counts that over
${\mathbf {C}}$
there are
$27$
lines, and over
${\mathbf {R}}$
the number of hyperbolic lines minus the number of elliptic lines is
$3$
. In general, the lines are defined over a field extension
$L$
and have an associated arithmetic type
$\alpha$
in
$L^*/(L^*)^2$
. There is an equality in the Grothendieck–Witt group
$\operatorname {GW}(k)$
of
$k$
,
\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]
where
$\operatorname {Tr}_{L/k}$
denotes the trace
$\operatorname {GW}(L) \to \operatorname {GW}(k)$
. Taking the rank and signature recovers the results over
${\mathbf {C}}$
and
${\mathbf {R}}$
. To do this, we develop an elementary theory of the Euler number in
$\mathbf {A}^1$
-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with one of those defined by the last-two named authors (using an extended Brill-Noether locus on a suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.
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