Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the pluricanonical section index${\it\delta}(V):=\text{min}\{m\mid P_{m}\geqslant 2\}$ since $1\leqslant {\it\delta}(V)\leqslant 18$ due to our previous series (I, II). Based on our further classification to 3-folds with ${\it\delta}(V)\geqslant 13$ and an intensive geometrical investigation to those with ${\it\delta}(V)\leqslant 12$, we prove that $\text{Vol}(V)\geqslant \frac{1}{1680}$ and that the pluricanonical map ${\rm\Phi}_{m}$ is birational for all $m\geqslant 61$, which greatly improves known results. An optimal birationality of ${\rm\Phi}_{m}$ for the case ${\it\delta}(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_{g}\geqslant 2$ in the last section.

We study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$. On the other hand, we identify them up to birational equivalence among all varieties of maximal Albanese dimension. We also describe the structure of varieties $X$ of maximal Albanese dimension, with holomorphic Euler characteristic $1$ and irregularity $2\dim X-1$.

The Nash problem asks about the existence of a correspondence between families of arcs through singularities of complex varieties and certain types of divisorial valuations. It has been positively settled in dimension 2 by Fernández de Bobadilla and Pe Pereira, and it was shown to have a negative answer in all dimensions ${\geq }4$ by Ishii and Kollár. In this note we discuss examples which show that the problem has a negative answer in dimension 3 as well. These examples also bring to light the different nature of the problem depending on whether it is formulated in the algebraic setting or in the analytic setting.

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed.

The examples considered come from particular special homaloidal and subhomaloidal linear systems, which are usually degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron.

The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of $\mathbf{P}^{m}$, with $3 \leq m \leq 5$. We get another series of examples by considering special cubic hypersurfaces through some surfaces in $\mathbf{P}^5$, or some 3-folds in $\mathbf{P}^7$ having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section of the paper.