Cindy De Volder, Antonio Laface
In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.
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We give a closed formula for the dimension of all linear systems in $\mathbb{P}^n$ with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree $n$. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.
We extend results of our previous papers, on ordinary multiple points of curves, and on the computation of their conductor, to ordinary multiple subvarieties of codimension one.
If $ D \in X$ is a curve with multiple points in a surface, a parabolic bundle defined on $(X;D)$ away from the singularities can be extended in several ways to a parabolic bundle on a resolution of singularities. We investigate the possible parabolic Chern classes for these extensions.
We show that there exists a unique possible definition, with certain natural properties, of the multiple point space of a holomorphic map between complex manifolds. Our construction coincides with the double point space and the k-th multiple point space for corank one map-germs, due to Mond. We also give some interesting properties of the double point space and prove that in many cases it can be computed as the zero locus of certain quotient of ideals.
This is the first of a two-part work on Kleiman's iterated multiple point spaces. We show general properties of these spaces, leading to explicit equations describing them for maps (of any corank) between complex manifolds. We also describe pathologies regarding dimension and lack of symmetry
We study the postulation of a general union $X\subset \mathbb {P}^3$ of one m-point $mP$ and $t$ disjoint lines. We prove that it has the expected Hilbert function, proving a conjecture by E. Carlini, M. V. Catalisano and A. V. Geramita.
Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.
Given a birational parameterization of an algebraic surface S in the projective space, the purpose of this paper is to investigate the sets of points on S whose preimage consists in k or more points, counting multiplicities. They are described explicitly in terms of Fitting ideals of some graded parts of the symmetric algebra associated to this parameterization.
Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $¶^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes Serre's $(S_1)$ property holds. Specifically, we prove that for any positive integers $h, e \ge 2$ with $(h,e) \neq (2,2)$ and $p \ge 5$ there is a homogeneous ideal $I$ in a polynomial ring over $K$ such that (1) the height of $I$ is $h$, (2) the Hilbert-Samuel multiplicity of…
Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been observed and reported on. All examples studied so far had just one fat point. In this note we introduce a new series of examples, which establishes for the first time the existence of unexpected hypersurfaces with multiple fat points. The key underlying idea is to study Fermat-type configurations of points in projective spaces.
We study Gromov-Witten invariants on the blow-up of P^n at a point, which is probably the simplest example of a variety whose moduli spaces of stable maps do not have the expected dimension. It is shown that many of these invariants can be interpreted geometrically on P^n as certain numbers of rational curves having a multiple point of given order at the blown up point. Moreover, all these invariants can actually be calculated, giving enumerative invariants of P^n which have not been known before.
We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $H\G$, where $H$ is a connected simple algebraic group embedded diagonally into $G := H^n$.
Non-linear geometry of multiple zeta values
本文探讨多重ζ值的非线性几何，通过热带几何、模空间、费曼积分等主题，提出基于行列式表示的几何框架，并提出未来研究问题。
Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given. The problem was already solved completely in the case that the polynomials have no multiple fixed points, in the author's previous papers. In this paper, we consider the case of having multiple fixed points, and obtain the formulae for generic unordered collections of holomorphic fixed point indices, for each given degree and for each given number of fixed p…
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