57 research outputs found
Keller–Osserman Phenomena for Kardar–Parisi–Zhang-Type Inequalities
No full textFor coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results for the noncoercive case, which had been previously established by the same author
A class of Osserman spaces
No full textWe prove that a symmetric space is Osserman if its complexification is a complex hyper-Kähler symmetric space. This includes all pseudo-hyper-Kähler as well as para-hyper-Kähler symmetric spaces. We extend the classification of pseudo-hyper-Kähler symmetric spaces obtained by the first and the third author to the class of para-hyper-Kähler symmetric spaces. These manifolds are possible targets for the scalars of rigid N=2 supersymmetric field theories with hypermultiplets on four-dimensional space-times with Euclidean signature
Keller–Osserman Phenomena for Kardar–Parisi–Zhang-Type Inequalities
No full textFor coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results for the noncoercive case, which had been previously established by the same author
Limit linear series in positive characteristic and Frobenius-unstable vector bundles on curves
No full textThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 243-248).(cont.) yield a new proof of a result of Mochizuki yield a new proof of a result of Mochizuki Frobenius-unstable bundles for C general, and hence obtaining a self-contained proof of the resulting formula for the degree of V₂.Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P¹ with ramification to order e[sub]i at general points P[sub]i the case that all e[sub]i are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in positive and mixed characteristics, yielding a result on lifting linear series from characteristic p to characteristic 0, and even showing promise for generalization to higher-dimensional varieties. Now, let C be a curve of genus 2 over a field k of positive characteristic, and V₂ the Verschiebung rational map induced by pullback under Frobenius on moduli spaces of semistable vector bundles of rank two and trivial determinant. We show that if the Frobenius-unstable vector bundles are deformation-free in a suitable sense, then they are precisely the undefined points of V₂, and may each be resolved by a single blow-up; in this setting, we are able to calculate the degree of V₂ in terms of the number of Frobenius-unstable bundles, and describe the image of the exceptional divisors. We finally examine the Frobenius-unstable bundles on C by studying connections with vanishing p-curvature on certain unstable bundles on C. Using explicit formulas for p-curvature, we completely describe the Frobenius-unstable bundles in characteristics 3, 5, 7. We classify logarithmic connections with vanishing p-curvature on vector bundles of rank 2 on P¹ in terms of self-maps of P¹ with prescribed ramification. Using our knowledge of such maps, we then glue the connections to a nodal curve and deform to a smooth curve toby Brian Osserman.Ph.D
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Linked determinantal loci and limit linear series
No full textWe study (a generalization of) the notion of linked determinantal loci recently
introduced by the second author, showing that as with classical determinantal loci, they
are Cohen-Macaulay whenever they have the expected codimension. We apply this to prove
Cohen-Macaulayness and flatness for moduli spaces of limit linear series, and to prove a
comparison result between the scheme structures of Eisenbud-Harris limit linear series and
the spaces of limit linear series recently constructed by the second author. This
comparison result is crucial in order to study the geometry of Brill-Noether loci via
degenerations
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Limit linear series moduli stacks in higher rank
No full textIn order to prove new existence results in Brill-Noether theory for rank-2 vector
bundles with fixed special determinant, we develop foundational definitions and results for
limit linear series of higher-rank vector bundles. These include two entirely new
constructions of "linked linear series" generalizing earlier work of the author for the
classical rank-1 case, as well as a new canonical stack structure for the previously
developed theory due to Eisenbud, Harris and Teixidor i Bigas. This last structure is new
even in the classical rank-1 case, and yields the first proper moduli space of
Eisenbud-Harris limit linear series for families of curves. We also develop results
comparing these three constructions
LIMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE
No full textWe introduce a notion of limit linear series for nodal curves which are not of
compact type. We give a construction of a moduli space of limit linear series, which works
also in smoothing families, and we prove a corresponding specialization result. For a more
restricted class of curves which simultaneously generalizes two-component curves and curves
of compact type, we give an equivalent definition of limit linear series, which is visibly
a generalization of the Eisenbud-Harris definition. Finally, for the same class of curves,
we prove a smoothing theorem which constitutes an improvement over known results even in
the compact-type case
A simple characteristic-free proof of the Brill-Noether theorem
No full textAbstract. We describe how the use of a different degeneration from that con-sidered by Eisenbud and Harris leads to a simple and characteristic-independent proof of the Brill-Noether theorem using limit linear series. As suggested by the degeneration, we prove an extended version of the theorem allowing for imposed ramification at up to two points. Although experts in the field have long been aware of the main ideas, we address some technical issues which arise in proving the full version of theorem. 1
DIMENSION COUNTS FOR LIMIT LINEAR SERIES ON CURVES NOT OF COMPACT TYPE
No full textWe first prove a generalized Brill-Noether theorem for linear series with
prescribed multivanishing sequences on smooth curves. We then apply this theorem to prove
that spaces of limit linear series have the expected dimension for a certain class of
curves not of compact type, whenever the gluing conditions in the definition of limit
linear series impose the maximal codimension. Finally, we investigate these gluing
conditions in specific families of curves, showing expected dimension in several cases,
each with different behavior. One of these families sheds new light on the work of Cools,
Draisma, Payne and Robeva in tropical Brill-Noether theory
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