Let $X$ be a general conic bundle over the projective plane with branch curve
of degree at least 19. We prove that there is no normal projective variety $Y$
that is birational to $X$ and such that some multiple of its anticanonical
divisor is effective. We also give such examples for 2-dimensional conic
bundles defined over a number field.
We show the existence of Bridgeland stability conditions on all Fano
threefolds, by proving a modified version of a conjecture by Bayer, Toda, and
the second author. The key technical ingredient is a strong Bogomolov
inequality, proved recently by Chunyi Li. Additionally, we prove the original
conjecture for some toric threefolds by using the toric Frobenius morphism.
Let $G$ be a reductive group over a field $k$ which is algebraically closed
of characteristic $p \neq 0$. We prove a structure theorem for a class of
subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As
applications, we derive semi-simplicity results, generalizing earlier results
of Serre proven in 1998, and also obtain an analogue of Luna's étale slice
theorem for suitable bounds on $p$.
In this article, we first describe codimension two regular foliations with
numerically trivial canonical class on complex projective manifolds whose
canonical class is not numerically effective. Building on a recent algebraicity
criterion for leaves of algebraic foliations, we then address regular
foliations of small rank with numerically trivial canonical class on complex
projective manifolds whose canonical class is pseudo-effective. Finally, we
confirm the generalized Bondal conjecture formulated by Beauville in some
special cases.
We show that for n ≥ 2 the space of closed n-cycles in a strongly (n − 2)-concave complex space has a natural structure of reduced complex space locally of finite dimension and represents the functor " analytic family of n-cycles " parametrized by Banach analytic sets.
We present several analogies between convex geometry and the theory of
holomorphic line bundles on smooth projective varieties or Kähler manifolds.
We study the relation between positive products and mixed volumes. We define
and study a Blaschke addition for divisor classes and mixed divisor classes,
and prove new geometric inequalities for divisor classes. We also reinterpret
several classical convex geometry results in the context of algebraic geometry:
the Alexandrov body construction is the convex geometry version of divisorial
Zariski decomposition; Minkowski's existence theorem is the convex geometry
version of the duality between the pseudo-effective cone of divisors and the
movable cone of curves.
By analogy with Green's Conjecture on syzygies of canonical curves, the
Prym-Green conjecture predicts that the resolution of a general level p
paracanonical curve of genus g is natural. The Prym-Green Conjecture is known
to hold in odd genus for almost all levels. Probabilistic arguments strongly
suggested that the conjecture might fail for level 2 and genus 8 or 16. In this
paper, we present three geometric proofs of the surprising failure of the
Prym-Green Conjecture in genus 8, hoping that the methods introduced here will
shed light on all the exceptions to the Prym-Green Conjecture for genera with
high divisibility by 2.
Let $X$ be a smooth projective variety over the complex numbers, and $\Delta
\subseteq X$ a reduced divisor with normal crossings. We present a slightly
simplified proof for the following theorem of Campana and Păun: If some
tensor power of the bundle $\Omega_X^1(\log \Delta)$ contains a subsheaf with
big determinant, then $(X, \Delta)$ is of log general type. This result is a
key step in the recent proof of Viehweg's hyperbolicity conjecture.
Haas' theorem describes all partchworkings of a given non-singular plane
tropical curve $C$ giving rise to a maximal real algebraic curve. The space of
such patchworkings is naturally a linear subspace $W_C$ of the
$\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow \Pi_C$ generated by the
bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of
this note is to provide an interpretation of affine subspaces of
$\overrightarrow \Pi_C $ parallel to $W_C$. To this purpose, we work in the
setting of abstract graphs rather than plane tropical curves. We introduce a
topological surface $S_\Gamma$ above a trivalent graph $\Gamma$, and consider a
suitable affine space $\Pi_\Gamma$ of real structures on $S_\Gamma$ compatible
with $\Gamma$. We characterise $W_\Gamma$ as the vector subspace of
$\overrightarrow \Pi_\Gamma$ whose associated involutions induce the same
action on $H_1(S_\Gamma,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this
statement another proof of Haas' original result.
In these notes we reformulate the classical Hilbert-Mumford criterion for GIT
stability in terms of algebraic stacks, this was independently done by
Halpern-Leinster. We also give a geometric condition that guarantees the
existence of separated coarse moduli spaces for the substack of stable objects.
This is then applied to construct coarse moduli spaces for torsors under
parahoric group schemes over curves.
Given a split semisimple group over a local field, we consider the maximal
Satake-Berkovich compactification of the corresponding Euclidean building. We
prove that it can be equivariantly identified with the compactification which
we get by embedding the building in the Berkovich analytic space associated to
the wonderful compactification of the group. The construction of this embedding
map is achieved over a general non-archimedean complete ground field. The
relationship between the structures at infinity, one coming from strata of the
wonderful compactification and the other from Bruhat-Tits buildings, is also
investigated.