14 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
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
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
Compactifications of moduli of points and lines in the projective plane
Projective duality identifies the moduli spaces and
parametrizing linearly general configurations of points
in and lines in the dual , respectively. The
space admits Kapranov's Chow quotient compactification
, studied also by Lafforgue, Hacking, Keel,
Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable
surfaces: it carries a family of certain reducible degenerations of
with "broken lines". Gerritzen and Piwek proposed a dual
perspective, a compact moduli space parametrizing certain reducible
degenerations of with smooth points. We investigate the
relation between these approaches, answering a question of Kapranov from 2003.Comment: 66 pages. Final version. To appear in International Mathematics
Research Notice
Point configurations, phylogenetic trees, and dissimilarity vectors
In 2004 Pachter and Speyer introduced the higher dissimilarity maps for
phylogenetic trees and asked two important questions about their relation to
the tropical Grassmannian. Multiple authors, using independent methods,
answered affirmatively the first of these questions, showing that dissimilarity
vectors lie on the tropical Grassmannian, but the second question, whether the
set of dissimilarity vectors forms a tropical subvariety, remained opened. We
resolve this question by showing that the tropical balancing condition fails.
However, by replacing the definition of the dissimilarity map with a weighted
variant, we show that weighted dissimilarity vectors form a tropical subvariety
of the tropical Grassmannian in exactly the way that Pachter--Speyer
envisioned. Moreover, we provide a geometric interpretation in terms of
configurations of points on rational normal curves and construct a finite
tropical basis that yields an explicit characterization of weighted
dissimilarity vectors.Comment: Final version. To appear in Proceedings of the National Academy of
Sciences of the United States of America (PNAS