We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$ of degree $d$ admits a stratification parameterized by graphs. We use it to determine the number of irreducible rational contact curves in $\mathbb{P}^{2n+1}$ with any Schubert condition. We give explicitely some of these invariants for $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$. We give another proof of the formula for the number of plane contact curves in $\mathbb{P}^{3}$ meeting the appropriate number of lines.
Trigonal curves provide an example of Brill-Noether special curves. Theorem 1.3 of [9] characterizes the Brill-Noether theory of general trigonal curves and the refined stratification by Brill-Noether splitting loci, which parametrize line bundles whose push forward to $\mathbb{P}^1$ has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill-Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected…
The aim of this paper is to generalize the $m-$Segre invariant for vector bundles to coherent systems. Let $X$ be a non-singular irreducible complex projective curve of genus $g$ over $\mathbb{C}$ and $(E,V)$ be a coherent system on $X$ of type $(n,d,k)$. For any pair of integers $m, t$, $0 < m < n$, $0 \leq t \leq k$ we define the $(m,t)-$Segre invariant, denoted by $S^α_{m,t}$ and show that $S^α_{m,t}$ induces a semicontinuous function on the families of coherent systems. Thus, $S^α_{m,t}$ gives a stratification of the moduli space $G(α;n,d,k)$ of $α-$stable coherent systems of type $(n,d,k)…
Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$. We prove that for general degree $k$ covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $W^r_d(C)$ for a general $k$-gonal curve (there are often components of different dimensions), extending results of Pflueger and Jensen-Ranganathan. The results here apply over any algeb…
We give a description of the $\mathrm{GL}_4$ Rapoport-Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl-Oort stratification. As an application of this, we also give a description of the supersingular locus of the Shimura variety for the group $\mathrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.
Let $S$ be a Puiseux series of the germ of an analytically irreducible plane curve $Z$. We provide a new perspective to construct a set of polynomials $F=\{F_1,\ldots, F_{g-1}\}$ associated to $S$, which is a special choice of maximal contact elements constructed in [AMB17] and approximate roots defined in [Dur18], [AM73a], [AM73b]. Using these polynomials as building blocks, we describe a set of generators of multiplier ideals of the form $\mathfrak{I}(αZ)$ with $0<α<1$ a rational number, which recovers the results about irreducible plane curves in [AMB17], [Dur18].
In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.
We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is isomorphic to the skeleton of the Berkovich analytification of the moduli space of algebraic spin curves, and give a geometric interpretation of the retraction map in terms of a tropicalization map.
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $ν$-gonal curve
Yeungnam University（岭南大学）；Universita’ degli Studi di Roma Tor Vergata（罗马托尔维加塔大学）
Let $X \subset \mathbb{P}^r$ be smooth and irreducible and for $k \ge 0$ let $ν_k(X)$ (resp., $δ_k(X)$) be the $k$-th contact (resp., the $k$-th secant) defect of $X$. For all $k \ge 0$ we have the inequality $ν_k(X) \ge δ_k(X)$ and in the case $k=1$ we characterize projective varieties $X$ for which equality holds, $\dim \mathrm{Sing}(X) \le δ_1(X) -1$ and the generic tangential contact locus is reducible.
Let X be a smooth irreducible complex projective curve of genus g > 1. In this paper, we give necessary and sufficient conditions for an unstable bundle of HN-lenght 2 to have a particular algebra of endomorphisms. Then, fixing the dimension of the algebra of endomorphisms we obtain a stratification of the moduli scheme such that each strata is a coarse moduli space. A particular case of interest is when the unstable bundles are simple. In that case the moduli space is fine. Topological properties of those moduli spaces are described and will depend on the generality of the curve X. Such resul…
With G=GL(n,C), let $\mathcal{X}_ΓG$ be the G-character variety of a given finitely presented group $Γ$, and let $\mathcal{X}^{irr}_ΓG \subset \mathcal{X}_ΓG$ be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E- polynomials of $\mathcal{X}_ΓG$ and the one for $\mathcal{X}^{irr}_ΓG$, generalizing a formula of Mozgovoy-Reineke [MR]. The proof uses a natural stratification of $\mathcal{X}_ΓG$ coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for…
A point $p\in\mathbb{P}^N$ of a projective space is $h$-identifiable, with respect to a variety $X\subset\mathbb{P}^N$, if it can be written as linear combination of $h$ elements of $X$ in a unique way. Identifiability is implied by conditions on the contact locus in $X$ of general linear spaces called non weak defectiveness and non tangential weak defectiveness. We give conditions ensuring non tangential weak defectiveness of an irreducible and non-degenerated projective variety $X\subset\mathbb{P}^N$, and we apply these results to Segre-Veronese varieties.
