342 research outputs found
The moduli space of hypersurfaces whose singular locus has high dimension
Let be an algebraically closed field and let and be integers with
and Consider the moduli space of
hypersurfaces in of fixed degree whose singular locus is
at least -dimensional. We prove that for large , has a unique
irreducible component of maximal dimension, consisting of the hypersurfaces
singular along a linear -dimensional subspace of . The proof
will involve a probabilistic counting argument over finite fields.Comment: Final version, including the incorporation of all comments by the
refere
Shapes of free resolutions over a local ring
We classify the possible shapes of minimal free resolutions over a regular
local ring. This illustrates the existence of free resolutions whose Betti
numbers behave in surprisingly pathological ways. We also give an asymptotic
characterization of the possible shapes of minimal free resolutions over
hypersurface rings. Our key new technique uses asymptotic arguments to study
formal Q-Betti sequences.Comment: 14 pages, 1 figure; v2: sections have been reorganized substantially
and exposition has been streamline
Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type
In this paper we show a partial answer the a question of C. Huneke and G.
Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded
countable Cohen-Macaulay representation type, and assume that R has an isolated
singularity. Is R then necessarily of graded finite Cohen-Macaulay
representation type? In particular, this question has an affirmative answer for
standard graded non-Gorenstein rings as well as for standard graded Gorenstein
rings of minimal multiplicity. Along the way, we obtain a partial
classification of graded Cohen-Macaulay rings of graded countable
Cohen-Macaulay type.Comment: 15 Page
Probing scattering phase shifts by attosecond streaking
Attosecond streaking is one of the most fundamental processes in attosecond
science allowing for a mapping of temporal (i.e. phase) information on the
energy domain. We show that on the single-particle level attosecond streaking
time shifts contain spectral phase information associated with the
Eisenbud-Wigner-Smith (EWS) time delay, provided the influence of the streaking
infrared field is properly accounted for. While the streaking phase shifts for
short-ranged potentials agree with the associated EWS delays, Coulomb
potentials require special care. We show that the interaction between the
outgoing electron and the combined Coulomb and IR laser fields lead to a
streaking phase shift that can be described classically
Alignment and algebraically special tensors in Lorentzian geometry
We develop a dimension-independent theory of alignment in Lorentzian
geometry, and apply it to the tensor classification problem for the Weyl and
Ricci tensors. First, we show that the alignment condition is equivalent to the
PND equation. In 4D, this recovers the usual Petrov types. For higher
dimensions, we prove that, in general, a Weyl tensor does not possess aligned
directions. We then go on to describe a number of additional algebraic types
for the various alignment configurations. For the case of second-order
symmetric (Ricci) tensors, we perform the classification by considering the
geometric properties of the corresponding alignment variety.Comment: 19 pages. Revised presentatio
Extension of formal conjugations between diffeomorphisms
We study the formal conjugacy properties of germs of complex analytic
diffeomorphisms defined in the neighborhood of the origin of .
More precisely, we are interested on the nature of formal conjugations along
the fixed points set. We prove that there are formally conjugated local
diffeomorphisms such that every formal conjugation
(i.e. ) does not extend to
the fixed points set of , meaning that it is not
transversally formal (or semi-convergent) along .
We focus on unfoldings of 1-dimensional tangent to the identity
diffeomorphisms. We identify the geometrical configurations preventing formal
conjugations to extend to the fixed points set: roughly speaking, either the
unperturbed fiber is singular or generic fibers contain multiple fixed points.Comment: 34 page
Frobenius Splittings
We give a gentle introduction to Frobenius splittings. Then we recall a few
results that have been obtained with the method.Comment: 21 pages, typos correcte
Heterotic Compactification, An Algorithmic Approach
We approach string phenomenology from the perspective of computational
algebraic geometry, by providing new and efficient techniques for proving
stability and calculating particle spectra in heterotic compactifications. This
is done in the context of complete intersection Calabi-Yau manifolds in a
single projective space where we classify positive monad bundles. Using a
combination of analytic methods and computer algebra we prove stability for all
such bundles and compute the complete particle spectrum, including gauge
singlets. In particular, we find that the number of anti-generations vanishes
for all our bundles and that the spectrum is manifestly moduli-dependent.Comment: 36 pages, Late
-prime and -primary -ideals on -schemes
Let be a flat finite-type group scheme over a scheme , and a
noetherian -scheme on which -acts. We define and study -prime and
-primary -ideals on and study their basic properties. In particular,
we prove the existence of minimal -primary decomposition and the
well-definedness of -associated -primes. We also prove a generalization
of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts
type theorem on graded rings for -regular and -rational properties.Comment: 54pages, added Example 6.16 and the reference [8]. The final versio
Non-commutative desingularization of determinantal varieties, I
We show that determinantal varieties defined by maximal minors of a generic
matrix have a non-commutative desingularization, in that we construct a maximal
Cohen-Macaulay module over such a variety whose endomorphism ring is
Cohen-Macaulay and has finite global dimension. In the case of the determinant
of a square matrix, this gives a non-commutative crepant resolution.Comment: 52 pages, 3 figures, all comments welcom
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