Plane Quartic Curve in Grassmannians and Dominance of the Restriction Map between Two Moduli Spaces

defined by sending E to E|C . It is shown in [4] that Φk is a dominant map to W,W and W , for k = 1, 2, 3, respectively. In this article, we give a proof of the dominance of the rational map Φ4. This is equivalent to the dominance of the rational map from M(3, 6) to SUC(2, 3KC) by twisting. For a general bundle E ∈ SUC(2, 3KC), we embed C with P2 into a Grassmannian Gr(5, 2) and take the pull-back of the universal quotient bundle of Gr(5, 2) to P2. This bundle is shown to be stable and have the Chern classes (3,6). As a quick consequence, we can obtain the old result that SUC(2,KC) is unirational since M(1, 4) is rational. Since the logarithmic bundles from the general arrangements of 6 lines on P2 form an open Zariski subset of M(3, 6), we expect to give an interpretation on the geometry of SUC(2,KC) in terms of the arrangement of 6 lines, using the dominance of the restriction map from M(1, 4).