Abstract
The Konno invariant of a projective variety X is the minimum geometric genus of the fiber of a rational pencil on X. It was computed by Konno for surfaces in \({\mathbf {P}}^3\), and in general can be viewed as a measure of the complexity of X. We estimate \(\mathrm{Konno}(X)\) for some natural classes of varieties, including sharp asymptotics for polarized K3 surfaces. In an appendix, we give a quick proof of a classical formula due to Deligne and Hoskin for the colength of an integrally closed ideal on a surface.
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Notes
Note Added in Proof: Nathan Chen has established the very interesting result that if \(A_d\) is a very general abelian surface with a polarization of type (1, d), then the degree of irrationality of \(A_d\) is \(\leqslant 4\). The argument proceeds by showing the the Kummer variety of A admits a two-fold rational covering of \({\mathbf {P}}^2\). Nonetheless, it seems to remain plausible that the degree of irrationality of a general K3 surface \(S_d\) goes to infinity with d.
In our setting, the vanishing in question is very elementary. In fact, the question being local, one can replace S by an affine neighborhood of x, so that \({\mathscr {O}}_{S^\prime }(-A)\) is globally generated. Choose a general section \(s \in {{\Gamma }} ( {\mathscr {O}}_{S^\prime }(-A))\) cutting out a curve \(\Gamma \subseteq S^\prime \). Then \(\Gamma \) is finite over S, so the vanishing of \(R^1\mu _*{\mathscr {O}}_{S^\prime }(-A)\) follows from the exact sequence \(0 \longrightarrow {\mathscr {O}}_{S^\prime } \longrightarrow {\mathscr {O}}_{S^\prime }(-A) \longrightarrow {\mathscr {O}}_{{\Gamma }}(-A) \longrightarrow 0\).
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Acknowledgements
We are grateful to Francesco Bastianelli, Nathan Chen, Craig Huneke, David Stapleton and Ruijie Yang for useful discussions.
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Research of the first author partially supported by NSF Grant DMS-1801870. Research of the second author partially supported by NSF Grant DMS-1739285.
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Ein, L., Lazarsfeld, R. The Konno invariant of some algebraic varieties. European Journal of Mathematics 6, 420–429 (2020). https://doi.org/10.1007/s40879-019-00322-x
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DOI: https://doi.org/10.1007/s40879-019-00322-x