In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$. We prove that if the Galois group has order at most $3$, it always extends to a subgroup of the Jonquières group associated with the point $P$. Conversely, with a degree of at least $4$, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$. In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.

Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$. For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$-orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$, which measures the growth rate of the heights of the points $f^{n}(P)$. Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode[STIX]{x1D6FC}_{f}(P)$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $X=\mathbb{P}^{4}$.

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

We exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

One is concerned with Cremona-like transformations, i.e., rational maps from $ P$n to $ P$m that are birational onto the image Y ⊂ $ P$m and, moreover, the inverse map from Y to $ P$n lifts to $ P$m. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.