Let $E$ be an elliptic curve. An irreducible algebraic curve $C$ embedded in $E^g$ is called weak-transverse if it is not contained in any proper algebraic subgroup of $E^g$, and transverse if it is not contained in any translate of such a subgroup.

Suppose $E$ and $C$ are defined over the algebraic numbers. First we prove that the algebraic points of a transverse curve $C$ that are close to the union of all algebraic subgroups of $E^g$ of codimension $2$ translated by points in a subgroup $\Gamma$ of $E^g$ of finite rank are a set of bounded height. The notion of closeness is defined using a height function. If $\Gamma$ is trivial, it is sufficient to suppose that $C$ is weak-transverse.

The core of the article is the introduction of a method to determine the finiteness of these sets. From a conjectural lower bound for the normalized height of a transverse curve $C$, we deduce that the sets above are finite. Such a lower bound exists for $g\le 3$.

Concerning the codimension of the algebraic subgroups, our results are best possible.

We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials $k\left[t\right]$ is not a diophantine subset of the field of rational functions $k\left(t\right)$ for many fields $k$.

In an earlier paper by one of us (Behrend), Donaldson–Thomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical $\mathbb{Z}$-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point.

Here we evaluate this invariant for the case of a singularity that is an isolated point of a ${ℂ}^{\ast }$-action and that admits a symmetric obstruction theory compatible with the ${ℂ}^{\ast }$-action. The answer is ${\left(-1\right)}^d$, where $d$ is the dimension of the Zariski tangent space.

We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of $n$ points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi–Yau threefold, we deduce that the Donaldson–Thomas invariant of the Hilbert scheme of $n$ points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande.

In a seminal work Lyubeznik [1997] introduces a category $F$-finite modules in order to show various finiteness results of local cohomology modules of a regular ring $R$ in positive characteristic. The key notion on which most of his arguments rely is that of a generator of an $F$-finite module. This may be viewed as an $R$ finitely generated representative for the generally nonfinitely generated local cohomology modules. In this paper we show that there is a functorial way to choose such an $R$-finitely generated representative, called the minimal root, thereby answering a question that was left open in Lyubeznik’s work. Indeed, we give an equivalence of categories between $F$-finite modules and a category of certain $R$-finitely generated modules with a certain Frobenius operation which we call minimal $\gamma$-sheaves.

As immediate applications we obtain a globalization result for the parameter test module of tight closure theory and a new interpretation of the generalized test ideals of Hara and Takagi [2004] which allows us to easily recover the rationality and discreteness results for $F$-thresholds of Blickle et al. [2008].