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"Geometry, Algebraic."
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Inflectionary Invariants for Isolated Complete Intersection Curve Singularities
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring
singular members. Let
eBook
Motivic Euler Products and Motivic Height Zeta Functions
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes,
in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the
study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant
compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a
topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic
behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above
moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools
for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of
Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with
exponentials constructed using Denef and Loeser’s motivic vanishing cycles.
eBook
Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models
We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic
threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball
quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of
the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a
detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space
of cubic surfaces is discussed in an appendix.
eBook
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.
eBook
Change and invariance : a textbook on algebraic insight into numbers and shapes
What is the connection between finding the amount of acid needed to reach the desired concentration of a chemical solution, checking divisibility by a two-digit prime number, and maintaining the perimeter of a polygon while reducing its area? The simple answer is the title of this book. The world is an interplay of variation and constancy -- a medley of differences and similarities -- and this change and invariance is, largely, a language of science and mathematics. This book proposes a unique approach for developing mathematical insight through the perspective of change and invariance as it applies to the properties of numbers and shapes.
Book
Deformation and Unobstructedness of Determinantal Schemes
A closed subscheme
First of all, we compute an upper
The
work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected
as conjectures in the final section.
eBook