19 research outputs found
On the cone of effective 2-cycles on
Fulton's question about effective -cycles on for
can be answered negatively by appropriately lifting to
the Keel-Vermeire divisors on . In
this paper we focus on the case of -cycles on , and we
prove that the -dimensional boundary strata together with the lifts of the
Keel-Vermeire divisors are not enough to generate the cone of effective
-cycles. We do this by providing examples of effective -cycles on
that cannot be written as an effective combination of the
aforementioned -cycles. These examples are inspired by a blow up
construction of Castravet and Tevelev.Comment: 22 pages, 4 figures. Final version. Minor corrections. To appear in
the European Journal of Mathematic
K3 surfaces with symplectic action
Let be a finite abelian group which acts symplectically on a K3 surface.
The N\'eron-Severi lattice of the projective K3 surfaces admitting
symplectic action and with minimal Picard number is computed by Garbagnati and
Sarti. We consider a -dimensional family of projective K3 surfaces with
symplectic action which do not fall in the above cases. If
is one of these K3 surfaces, then it arises as the minimal resolution of a
specific -cover of branched along six general
lines. We show that the N\'eron-Severi lattice of with minimal Picard
number is generated by smooth rational curves, and that specializes to
the Kummer surface . We relate to the K3
surfaces given by the minimal resolution of the -cover of
branched along six general lines, and the corresponding
Hirzebruch-Kummer covering of exponent of .Comment: 24 pages, 6 figures. Final version with minor corrections and
additions. To appear in the Rocky Mountain Journal of Mathematic
KSBA compactification of the moduli space of K3 surfaces with purely non-symplectic automorphism of order four
We describe a compactification by KSBA stable pairs of the five-dimensional
moduli space of K3 surfaces with purely non-symplectic automorphism of order
four and lattice polarization. These K3 surfaces can
be realized as the minimal resolution of the double cover of
branched along a specific curve. We
show that, up to a finite group action, this stable pairs compactification is
isomorphic to Kirwan's partial desingularization of the GIT quotient
with the symmetric linearization.Comment: 26 pages, 6 figures. We explain the connection with Alexeev-Thompson
work on ADE surfaces. Comments are welcom
Decomposition of Lagrangian classes on K3 surfaces
We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.https://arxiv.org/abs/2001.00202Othe
A Pascal's theorem for rational normal curves
Pascal's Theorem gives a synthetic geometric condition for six points
in to lie on a conic. Namely, that the intersection
points , ,
are aligned. One could ask an analogous
question in higher dimension: is there a coordinate-free condition for
points in to lie on a degree rational normal curve? In this
paper we find many of these conditions by writing in the Grassmann-Cayley
algebra the defining equations of the parameter space of ordered points
in that lie on a rational normal curve. These equations were
introduced and studied in a previous joint work of the authors with
Giansiracusa and Moon. We conclude with an application in the case of seven
points on a twisted cubic.Comment: 16 pages, 1 figure. Comments are welcom
Families of pointed toric varieties and degenerations
The Losev-Manin moduli space parametrizes pointed chains of projective lines.
In this paper we study a possible generalization to families of pointed
degenerate toric varieties. Geometric properties of these families, such as
flatness and reducedness of the fibers, are explored via a combinatorial
characterization. We show that such families are described by a specific type
of polytope fibration which generalizes the twisted Cayley sums, originally
introduced to characterize elementary extremal contractions of fiber type
associated to projective -factorial toric varieties with positive
dual defect. The case of a one-dimensional simplex can be viewed as an
alternative construction of the permutohedra.Comment: 20 pages, 5 figures. Comments are welcom
Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces
It is known that some GIT compactifications associated to moduli spaces of
either points in the projective line or cubic surfaces are isomorphic to
Baily-Borel compactifications of appropriate ball quotients. In this paper, we
show that their respective toroidal compactifications are isomorphic to moduli
spaces of stable pairs as defined in the context of the MMP. Moreover, we give
a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case
of eight labeled points in the projective line.Comment: 35 pages. Comments are welcom
The non-degeneracy invariant of Brandhorst and Shimada families of Enriques surfaces
Brandhorst and Shimada described a large class of Enriques surfaces, called
-generic, for which they gave generators for the
automorphism groups and calculated the elliptic fibrations and the smooth
rational curves up to automorphisms. In the present paper, we give lower bounds
for the non-degeneracy invariant of such Enriques surfaces, we show that in
most cases the invariant has generic value , and we present the first known
example of complex Enriques surface with infinite automorphism group and
non-degeneracy invariant not equal to .Comment: 22 pages, 2 figures. Comments welcome
Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces
Smooth minimal surfaces of general type with , , and
constitute a fundamental example in the geography of algebraic surfaces, and
the 28-dimensional moduli space of their canonical models admits a
modular compactification via the minimal model program. We
describe eight new irreducible boundary divisors in such compactification
parametrizing reducible stable surfaces. Additionally, we study the relation
with the GIT compactification of and the Hodge theory of the
degenerate surfaces that the eight divisors parametrize.Comment: 40 pages. Comments are welcom