The object of study is the moduli space \(\mathrm{Mor}(C,X,y)\) of maps from a fixed (smooth projective geometrically irreducible) curve \(C\) to a variety \(X\) such that the cycle of its image is some fixed element \(y\) of \(\mathrm{NS}(X)^\vee\). There is known an explicit lower bound on the dimension of the irreducible components of \(\mathrm{Mor}(C,X,y)\), depending on the dimension of \(X\), the genus of \(C\) and the intersection pairing of \(y\) with \(K_X\). This bound is called the expected dimension of \(\mathrm{Mor}(C,X,y)\). In this context one can ask for the actual dimension of this space and the number of its irreducible components.
In this article, the author answers these questions in a specific situation. There, \(X\) should be a smooth projective geometrically irreducible variety defined over \(\mathbb{Q}\), whose Cox ring is given as a quotient of a polynomial ring \(k[x_1,\ldots,x_n]\) by one equation \(f\). Moreover, \(f\) should be linear with respect to a subset of the variables \(x_i\) in such a way that the coefficients of these variables are pairwise coprime monomials in the remaining variables.
There are more explicit combinatorial restrictions on \(f\) and \(y \in \mathrm{Eff}(X)^\vee\), which have to be fulfilled. But then the main theorem states, that if \(C\) is defined over \(\mathbb{Q}\), then there is an open dense irreducible subset \(\mathrm{Mor}(C,X,y)^\circ\) of \(\mathrm{Mor}(C,X,y)\) that has the expected dimension. This open subset consists of all maps which do not factor through the divisors of sections corresponding to the variables \(x_i\) in the Cox ring.
Since \(C\) and \(X\) are defined over \(\mathbb{Q}\), the theorem can be established by counting the number of points in reductions of open subsets of \(\mathrm{Mor}(C,X,y)\) modulo a prime \(p\). The actual dimension can then be deduced from the asymptocial behaviour of the number of \(\mathbb{F}_{p^r}\)-points for \(r \to \infty\).
Finally, the author gives two classes of examples. On the one hand, the assumptions are fulfilled for so-called intrinsic quadrics, see also [D. Bourqui, Manuscr. Math. 135, No. 1–2, 1–41 (2011; Zbl 1244.14018)]. On the other hand, minimal resolutions of singular del Pezzo surfaces are considered. By the classification found in [U. Derenthal, “Singular del Pezzo surfaces whose universal torsors are hypersurfaces”, arXiv:math.AG/0604194], there are \(20\) such surfaces which fulfill the assumptions for at least one non-zero \(y\). Actually, \(y\) can be chosen arbitrarily for ten of those surfaces. For the remaining ten, only a subcone of \(\mathrm{Eff}(X)^\vee\) satisfies the assumptions of the theorem. The author gives the ratios of the volumes of this subcone and \(\mathrm{Eff}(X)^\vee\).