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  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 $8,9,10$. (Added 11 June 2021: this paper has now appeared on the arXiv.) 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  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 , 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  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  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.

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

 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.

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

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

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