Luis Renato G. Dias, Susumu Tanabé, Mihai Tibar
We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map $\bR^n\to\bR^p$ in case $p>1$. We describe an effective estimation of the "nontrivial" part of the bifurcation locus.
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We show that the number of bifurcation points at infinity of a polynomial function f : C2 -> C is at most the number of branches at infinity of a generic fiber of f and that this upper bound can be diminished by one in certain cases.
We show that the variation of the topology at infinity of a two-variable polynomial function is localisable at a finite number of "atypical points" at infinity. We construct an effective algorithm with low complexity in order to detect sharply the bifurcation values of the polynomial function.
Bifurcations of polynomial maps with diffeomorphic and connected fibers
本文构造并证明了存在带正则分歧值的多项式映射，且该值邻域内的纤维保持连通并相互微分同胚。
Real regulator maps with finite 0-locus
研究双变量拉格朗日多项式在条件下的调节映射，证明其扩展的调节函数和A多项式位势均为有限集。
Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…
In this paper, we study maps between moduli spaces of lattice-polarized K3 surfaces induced by sublattices of prime index. We show that these maps can be used to determine if a rational point of the moduli space belongs to the Noether-Lefschetz locus. As an application, we prove that the Bombieri-Lang conjecture implies non-density statements for the rational points in the Noether-Lefschetz locus, as predicted by a conjecture of Shafarevich.
Parabolic bifurcation loci in the spaces of rational functions
本文给出了 $\mathbb{P}^1$ 上次数 $d>1$ 的有理函数空间 $\operatorname{Rat}_d$ 中抛物分歧轨迹的几何描述，推广了首一多项式情形的研究，且结果对二次有理函数同样具有创新性。
We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series.
We prove that the morphism that maps a rational ruled surface to its singular locus is genericaly injective modulo isomophism and duality. We also calculate the dimension and the degre of its image.
We study the surjectivity of suitable weighted Gaussian maps which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus of curves endowed with a higher root of the canonical bundle having associated linear system of dimension at least r + 1. In particular, we get a bound on the dimension of its Zariski tangent space, which turns out to be sharp in the special case r = 0. Finally, we describe this locus in the case of complete intersection curves.
Probability-theoretic interpretation of degeneracy locus formulas
本文用随机有理六顶点模型解释退化轨迹的Schwartz-MacPherson类，完成三种经典类型的稳定SSM理论，将斜类与对称类分别解释为概率和Chebyshev矩。
Higher Gaussian maps on the hyperelliptic locus and second fundamental form
本文研究超椭圆曲线canonical bundle的高阶偶数Gaussian映射，确定其秩并描述其核。利用此结果分析超椭圆Torelli映射j_h及其二阶基本形式，研究其切空间中的isotropic子空间。
In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $\operatorname{Rat}_d^N$ of degree $d$ rational maps $\mathbb{P}^N\to\mathbb{P}^N$ modulo the conjugation action of $\operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $N\ge3$ or $d\ge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $\operatorname{Rat}_2^2$.
We consider two varieties associated to a web of quadrics W in the projective space of dimension 7. One is the base locus and the second one is the double cover of the three dimensional projective space branched along the determinant surface of W. We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W contains a plane then in the generic case the two varieties are birational.
We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.
Let $k$ be an algebraically closed field of characteristic $0$ and $\mathcal{M}_d$ the moduli space of rational maps on $\mathbb{P}^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\subset \mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus $\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[ϕ]\in\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch-Popp-Oort characterization of singular points on the…
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