2,166 research outputs found
Machine learning detects terminal singularities
Algebraic varieties are the geometric shapes defined by systems of polynomial
equations; they are ubiquitous across mathematics and science. Amongst these
algebraic varieties are Q-Fano varieties: positively curved shapes which have
Q-factorial terminal singularities. Q-Fano varieties are of fundamental
importance in geometry as they are "atomic pieces" of more complex shapes - the
process of breaking a shape into simpler pieces in this sense is called the
Minimal Model Programme. Despite their importance, the classification of Q-Fano
varieties remains unknown. In this paper we demonstrate that machine learning
can be used to understand this classification. We focus on 8-dimensional
positively-curved algebraic varieties that have toric symmetry and Picard rank
2, and develop a neural network classifier that predicts with 95% accuracy
whether or not such an algebraic variety is Q-Fano. We use this to give a first
sketch of the landscape of Q-Fanos in dimension 8. How the neural network is
able to detect Q-Fano varieties with such accuracy remains mysterious, and
hints at some deep mathematical theory waiting to be uncovered. Furthermore,
when visualised using the quantum period, an invariant that has played an
important role in recent theoretical developments, we observe that the
classification as revealed by ML appears to fall within a bounded region, and
is stratified by the Fano index. This suggests that it may be possible to state
and prove conjectures on completeness in the future. Inspired by the ML
analysis, we formulate and prove a new global combinatorial criterion for a
positively curved toric variety of Picard rank 2 to have terminal
singularities. Together with the first sketch of the landscape of Q-Fanos in
higher dimensions, this gives new evidence that machine learning can be an
essential tool in developing mathematical conjectures and accelerating
theoretical discovery.Comment: 20 pages, 11 figures, 3 table
On Fano and Calabi-Yau varieties with hypersurface Cox rings
This thesis contributes to the explicit classification
of Fano and Calabi-Yau varieties.
First, we deal with complete intersections in
projective toric varieties that arise from a
non-degenerate system of Laurent polynomials.
Here we obtain Bertini type statements on canonical and
terminal singularities. This enables us to classify
all non-toric terminal Fano threefolds that
arise as a general complete intersection in a fake weighted projective space.
The second chapter is devoted to
the classification of all smooth Fano fourfolds of Picard number two
that have a general hypersurface Cox ring.
Using the Cox ring based description of these varieties
we investigate their birational geometry and compute Hodge numbers.
Moreover, we present a toolbox for constructing examples of general
hypersurface Cox rings including several factoriality criteria for
graded hypersurface rings.
Finally, we give classification results on smooth Calabi-Yau threefolds
of Picard number one and two that have a general hypersurface Cox ring
Toward the classification of higher-dimensional toric Fano varieties
The purpose of this paper is to give basic tools for the classification of
nonsingular toric Fano varieties by means of the notions of primitive
collections and primitive relations due to Batyrev. By using them we can easily
deal with equivariant blow-ups and blow-downs, and get an easy criterion to
determine whether a given nonsingular toric variety is a Fano variety or not.
As applications of these results, we get a toric version of a theorem of Mori,
and can classify, in principle, all nonsingular toric Fano varieties obtained
from a given nonsingular toric Fano variety by finite successions of
equivariant blow-ups and blow-downs through nonsingular toric Fano varieties.
Especially, we get a new method for the classification of nonsingular toric
Fano varieties of dimension at most four. These methods are extended to the
case of Gorenstein toric Fano varieties endowed with natural resolutions of
singularities. Especially, we easily get a new method for the classification of
Gorenstein toric Fano surfaces.Comment: 36 pages, Latex2e, to appear in Tohoku Math.
Gorenstein toric Fano varieties
We investigate Gorenstein toric Fano varieties by combinatorial methods using
the notion of a reflexive polytope which appeared in connection to mirror
symmetry. The paper contains generalisations of tools and previously known
results for nonsingular toric Fano varieties. As applications we obtain new
classification results, bounds of invariants and formulate conjectures
concerning combinatorial and geometrical properties of reflexive polytopes.Comment: AMS-LaTeX, 29 pages with 5 figure
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