AbstractWe define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $$C^{\infty }(N\times [0,1],P)$$
C
∞
(
N
×
[
0
,
1
]
,
P
)
is dense if and only if the pair of dimensions $$(\dim N, \dim P)$$
(
dim
N
,
dim
P
)
is in the extra-nice dimensions. This result is parallel to Mather’s characterization of the nice dimensions as the pairs (n, p) for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $${\mathscr {A}}_e$$
A
e
-codimension 1 hyperplane sections. They are also related to the simplicity of $${\mathscr {A}}_e$$
A
e
-codimension 2 germs. We give a sufficient condition for any $${\mathscr {A}}_e$$
A
e
-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.
AbstractA great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.
Abstract
The moduli space
M
¯
0,
n
(
ℙ
1
,
1
)
${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$
of n-pointed stable maps is a Mori dream space whenever the moduli space
M
¯
0
,
n
+
3
of
(
n
+
3
)
${{\bar{M}}_{0,n+3}}\; \text{of} \;(n+3)$
pointed rational curves is, and
M
¯
0
,
n
(
ℙ
1
,
1
)
${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$
is a log Fano variety for n ≤ 5.
In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.
Abstract
We prove a generalisation of the Brill-Noether theorem for the variety of special divisors
$W^r_d(C)$
on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of
$W^r_d(C)$
. We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus
$1$
curves to arbitrary genus.
All mammals have a well-developed hippocampus compared to that of fish, reptiles, and birds, although the latter still have homologous structures. The cells of the hippocampus have differentiated roles: place cells become active and rearrange themselves in new environments, which create new and stable maps of those environments. Grid cells are able to approximate distances, forming an additional neuronal basis for spatial navigation. The hippocampus and olfactory bulbs have intimately related functions. The story of patient H.M. revealed that declarative memories are consolidated by the hippocampus, but procedural memories can be established without hippocampal involvement. Declarative memories remain vulnerable to disruption and forgetting up to about 3 years after memorization. Memories consolidated during sleep are less prone to interference and more stable than memories followed by additional stimulation or learning.
The extra-nice dimensions
