$ \newcommand\A{\mathrm{A}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} $

Asher Auel and I have discussed the following definition, which is fun to think about.

Jonality. Let $X$ be an algebraic variety over a field $k$. The jonality of $X$ is the smallest $g$ such that there exists a smooth proper geometrically connected curve $C$ of genus $g$ defined over $k$ and a non-constant morphism $C\rightarrow X$.

Jonality is pronounced as in “Johan” plus “gonality”.

The jonality of a curve of genus $g$ is … $g$.

The jonality of projective space $\bP^r$ is $0$ since $\bP^r$ contains lines.

The jonality of a degree $d$ hypersurface $X$ in $\bP^r$ is at most the jonality of a degree $d$ smooth plane curve, i.e., $\frac{(d-1)(d-2)}{2}$ since we can intersect $X$ with $(r-2)$ generic hyperplanes. But, the jonality is typically smaller: think of the lines in a cubic surface.

If $A$ is a simple abelian $g$-fold, its jonality is at least $g$. If additionally $A$ is not isogeneous to a Jacobian, then the jonality is at least $g+1$.

The jonality of K3 surfaces is interesting: apparently it is zero (at least over $\bC$).

The previous examples were geometric in nature. The jonality is also an interesting arithmetic invariant. Asher Auel and I are working on the open problem of whether every Severi–Brauer variety $X$ defined over a field $k$ contains a (possibly singular genus $1$ curve). In other words, we are asking whether the jonality of $X$ is at most $1$. This was a question asked, in different terms, by Pete Clark and David Saltman.

Let $D$ be a division algebra of degree $d$ and let $X$ be its Severi–Brauer variety, an étale-twisted form of $\bP^{d-1}$. If $d=2$, then $X$ is a genus $0$ curve so its jonality is $0$. If $d\geq 3$, then $X$ contains no maps from any (geometrically) rational curve, so its jonality is at least $1$.

If $d=3,4,5$, then Johan de Jong and Wei Ho showed [3] that there is a genus $1$ curve mapping to $X$; i.e., the jonality is $1$ in these cases. Auel has proved the same result for $d=6$ using cohomological techniques to combine the cases of $d=2,3$.

Above $d=6$, little is known in general. Recently, with an observation of Saltman, Auel and I proved the following theorem.

Theorem. If $D$ is a division algebra of degree $7$ over a global field of characteristic prime to $7$, then $D$ is split by a genus $1$ curve; i.e., the jonality of the Severi–Brauer variety of $D$ is $1$.

This result will appear in forthcoming work which also gives new examples in degrees $8,9,10$. In the meantime, I would be very interested to hear of other arithmetically interesting cases of jonality computations.

Added 04 March 2021: Asher has done some more digging and has uncovered the following additional facts. The jonality has been extensively studied for abelian varieties.

  • The jonality of abelian varieties is studied in a paper [1] of Bardelli, Ciliberto, and Verra. They write $\gamma(X)$ for the jonality of $X$ and prove some results for general abelian varieties. Their best results in all dimensions are however superseded by a result of Pirola [4], which says that if $A$ is a very general abelian $g$-fold over $\bC$, then $\gamma(A)>\frac{g(g+1)}{2}$.
  • The gonality of an algebraic variety $X$ is defined in [2] by Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery to be the minimum of the gonalities of all normalizations of irreducible proper curves in $X$. Since the (geometric) gonality of a genus $0$ curve is $0$, that of a genus $1$ curve is $2$, and that of a genus $2$ curve is $2$ (as they are hyperelliptic), bounds on the gonality of $X$ have implications for the jonality as well. However, plausibly $X$ might contain high-genus hyperelliptic curves and hence have large jonality but small gonality. For instance, Voisin proves [5] that the gonality of the very general abelian $g$-fold over $\bC$ is linear in $g$; specifically, if $g\geq 2k-1$, then the gonality is at least $k+1$.
  • Using bend and break techniques, if $X$ has non-nef canonical bundle, then there are rational curves on $X$ (at least geometrically). So, in this case the jonality is $0$. See for example Section 8 of the lecture notes of Debarre.

References

[1] Bardelli, Ciliberto, Verra, Curves of minimal genus on a general abelian variety, Compositio Math. 96 (1995), no. 2, 115–147.

[2] Bastianelli, De Poi, Ein, Lazarsfeld, Ullery, Measures of irrationality for hypersurfaces of large degree, Compos. Math. 153 (2017), no. 11, 2368-2393. arXiv:1511.01359.

[3] de Jong and Ho, Genus one curves and Brauer–Severi varieties, Math. Res. Lett. 19 (2012), no. 6, 1357-1359. arXiv:1207.4810.

[4] Pirola, Abel-Jacobi invariant and curves on generic abelian varieties, Abelian varieties (Egloffstein, 1993), 237–249, de Gruyter, Berlin, 1995.

[5] Voisin, Chow ring and gonality of general abelian varieties, Ann. H. Lebesgue 1 (2018), 313–332. arXiv:1802.07153.