A note on Kawaguchi–Silverman conjecture

We collect some results on endomorphisms on projective varieties related to the Kawaguchi–Silverman conjecture. We discuss certain conditions on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi–Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.

On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

Homological Projective Duality for Linear Systems with Base Locus

We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a homological projective (HP) dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result also holds true when $Y$ is a noncommutative variety or just a category. We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

Cremona transformations and derived equivalences of K3 surfaces

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.