Andrea Bruno, Edoardo Sernesi
Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $ρ(g,r,n)$ is non-negative every component of the Petri locus $P^r_{g,n}\subset \M_g$ whose general member is a curve $C$ such that $W^{r+1}_n(C) = \emptyset$, has codimension one in $\M_g$.
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Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.
Two-pointed Prym-Brill-Noether Loci and coupled Prym-Petri theorem
本文建立了具有特殊消没的双点Prym-Brill-Noether轨迹，利用D型vexillary退化轨迹的K-理论公式确定其K-理论类，并借助耦合Prym-Petri映射证明了双点Prym-Petri定理，推广了前人的无点和带点情形。
Higher order Petri Loci
本文研究高阶皮特里利域的结构，证明其存在 codimension k 的组件。
We analyze the stratification of the moduli space S_g of spin curves of genus g given by the dimension of the theta-characteristic. Using the relation between gaussian maps and the strata S_g^r, we construct "regular" components of S_g^r having expected codimension r(r+1)/2 inside S_g. We also relate moduli spaces of pointed curves with a moving spin structure to the classical Gieseker-Petri loci in M_g. We show that the locus of curves for which the Gieseker-Petri theorem fails for a pencil is always a divisor on M_g. Finally, we give a sufficient criterion for the injectivity of Gaussian map…
In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.
We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.
We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym-Petri map and prove a pointed version of the Prym-Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De Concini-Pragacz on the unpointed case. Finally, we show that Prym varieties are Prym-Tyurin varietie…
Let C be a smooth projective complex curve of genus $g\geq2$. We investigate the Brill-Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least $k$ independent sections. Using the Hecke correpondence we construct a fundamental class, which determines the non-emptiness of this locus at least when $C$ is a Petri curve. We prove that in many expected cases the Brill-Noether locus is non-empty. For some values of $k$ the result is best possible.
We provide a bound on the $Θ$-regularity of an arbitrary reduced and irreducible curve embedded in a polarized abelian variety in terms of its degree and codimension. This is an "abelian" version of Gruson-Lazarsfeld-Peskine's bound on the Castelnuovo--Mumford regularity of a non-degenerate curve embedded in a projective space. As an application, we provide a Castelnuovo type bound for the genus of a curve in a (non necessarily principally) polarized abelian variety. Finally, we bound the $Θ$-regularity of a class of higher dimensional subvarieties in Jacobian varieties, i.e. the Brill-Noether…
In this short note we show that, for any ample embedding of a variety of dimension at least two in a projective space, all high enough degree Veronese re-embeddings have non-empty Terracini loci.
The point of this short note concerns with two facts on the scheme of secant loci. The first one is an attempt to describe the tangent cone of these schemes globally and the second one is a comparision on the dimension of the tangent spaces of various schemes of secant loci.
We determine non hyper elliptic curves of genus $g(C)\geq 9$, such that for some very ample line bundle on them and for some integers d and r with some prescribed assumptions, the dimension of secant loci, attains one less than its maximum value. Then we proceed to generalize and extend a problem of M. Coppens to Secant Loci.
It is known that the fundamental groups of smooth loci of Log del Pezzo Surfaces are finite groups. The aim of this note is to study these finite groups. A short table containing these groups is given. And lots of groups on the table are proved to be fundamental groups.
Brill-Noether loci
本文通过研究椭圆曲线链邻域中的非奇异曲线，重新证明了 Brill-Noether 轨迹在特定余维数范围内的非空性及 Petri 映射的满射性，并推广了同余维数轨迹互异性的已知结果。
The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smo…
Local structure of theta divisors and related loci of generic curves
本文研究一般曲线 theta 除子及 Brill-Noether 轨迹的局部结构，证明在 Petri 映射单射条件下轨迹局部 étale 同构于切锥，并由此解决单值性猜想等问题。
Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formul…
In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls.
A note on the Brill-Noether loci of small codimension in moduli space of stable bundles
本文研究了稳定束模空间中Brill-Noether亚变种的小余维数性质，证明了不同亏格情况下该亚变种的理性质和Chow群平凡性。
Given a curve $C$ and a linear system $\ell$ on $C$, the secant locus $V_e^{e-f}( \ell )$ parametrises effective divisors of degree $e$ which impose at most $e-f$ conditions on $\ell$. For $E \to C$ a vector bundle of rank $r$, we define determinantal subschemes $H_e^{e-f} ( \ell ) \subseteq \mathrm{Hilb}^e ( \mathbb{P} E )$ and $Q_e^{e-f} (V) \subseteq \mathrm{Quot}^{0, e} ( E^* )$ which generalise $V_e^{e-f} ( \ell )$, giving several examples. We describe the Zariski tangent spaces of $Q_e^{e-f} (V)$, and give examples showing that smoothness of $Q_e^{e-f} (V)$ is not necessarily controlled …
Let $π: X \longrightarrow C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback $π^*E$ is a $H-$stable bundle over $X$. This result allows us to relate the corresponding moduli spaces of stable bundles $\mathcal{M}_C(2,d)$ and $\mathcal{M}_{X,H}(2,df,0)$ through an injective morphism. We study the induced morphism at the level of Brill-Noether loci to construct examples of Brill-Noether loci on fibered surfaces. Results concerning the emptiness of Brill-Noether …
Erratum to the paper: Asymptotic Invariants of Base Loci
本文针对论文《Asymptotic Invariants of Base Loci》中一项技术结果的证明漏洞进行勘误，并提供正确证明。
Brill-Noether loci on an Enriques surface covered by a Jacobian Kummer surface
针对被 Jacobian Kummer 曲面覆盖的 Enriques 曲面，本文证明其特定秩 2 稳定向量丛模空间中存在真第一 Brill-Noether 轨迹。
In this note we give a rational uniformisation equation of the discriminant loci associated to a non-degenerate affine complete intersection variety. To show this formula we establish a relation of the fibre-integral with the hypergeometric function of Horn and that of Gel'fand-Kapranov-Zelevinski.
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
We produce open subsets of the moduli space of metric graphs without separating edges where the dimensions of Brill-Noether loci are larger than the corresponding Brill-Noether numbers. These graphs also have minimal rank determining sets that are larger than expected, giving couterexamples to a conjecture of Luo. Furthermore, limits of these graphs have Brill-Noether loci of the expected dimension, so dimensions of Brill-Noether loci of metric graphs do not vary upper semicontinuously in families. Motivated by these examples, we study a notion of rank for the Brill-Noether locus of a metric g…
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…
This note is an appendix to 'Measures of irrationality for hypersurfaces of large degree' by L. Ein, R. Lazarsfeld and B. Ullery. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in $\mathbb{P}^{n+1}$. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d \geq 3n-2 \geq 7$, and a dominant rational map $f \colon X \dashrightarrow \mathbb{…
In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n-2, 2) with n>3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type Shimura variety and its lift in the integral model by certain homogeneous spaces. As applications, we provide a mass formula for the superspecial locus.
We collect some results on endomorphisms on projective varieties related with the Kawaguchi-Silverman conjecture. We discuss certain condition on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi-Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.
Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigo…
Note sur le lieu discriminant d'une hypersurface de bi-degré $(m, n).$
本文提出一种基于群胚描述单值性的新拓扑方法，用于研究射影空间乘积中代数簇的判别轨迹，并以双次数 $(m, n)$ 超曲面为例进行说明。
In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.
In this note, we study Shimura varieties for the groups $\mathrm{GU}(V)$, where $V$ is a Hermitian space relative to a CM extension $E/E^+$. We give a description of the supersingular locus of the fiber at a prime $ν$ over $p$ of such a Shimura variety, under the assumptions that $\mathrm{dim}_E(V) \leq 4$ and that the prime $p$ splits completely in $E^+$ and is unramified in $E$.
Let $\mathcal H_g$ be the moduli space of genus $g$ hyperelliptic curves. In this note, we study the locus $\mathcal L$ in $\mathcal H_g$ of curves admitting a $G$-action of given ramification type $σ$ and inclusions between such loci. For each genus we determine the list of all possible groups, the inclusions among the loci, and the corresponding equations of the generic curve in $\mathcal L$. The proof of the results is based solely on representations of finite subgroups of $PGL_2 (\mathbb C)$ and the Riemann-Hurwitz formula.
Quasimaps to Nakajima varieties as critical loci
研究Nakajima箭图簇X上从ℙ¹到X且将无穷远点映射到给定点的拟映射空间，发现其可整体表示为显式函数的临界轨迹。
We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier--Mukai transform, where we recall that the structure group $\mathrm{GL}(n,\mathbb{C})$ is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. …
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
In this note we use an example of Mukai to construct semistable bundles of rank 3 with 6 independent sections on a curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill-Noether loci with negative expected dimension.
For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standar…
A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of regular components of M_g(Y,b) when Y is a product of projective spaces. This result can also be seen as a general position theorem for Brill-Noether loci in M_g.
This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.
Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes
本文提出通过热带几何定义等仿不变量，并研究平面凸域在特定条件下的极限结构及性质。
We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. …
On the GKZ discriminant locus
本文证明GKZ判别 locus 是所有判别 locus 的直接并集，无需取闭包，回答了Kite和Segal的问题。
In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $Ω(\log(N)…
In this note, we introduce and study position space Feynman quadrics that are the loci of divergences of the position space Feynman integrals for Euclidean massless scalar quantum field theories. We prove that the Feynman quadrics define objects in the category of mixed Tate motives for complete graphs with a bound on the number of vertices. This result shows a strong contrast with the graph hypersurfaces approach which produces also non-mixed Tate examples.
The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\mathbb{C}^*)^2$, the complex algebraic plane curves whose coamoebas are of maximal area (counted with multiplicity) are defined over $\mathbb{R}$, and their real loci are Harnack curves possibly with ordinary real isolated double points (c.f. \cite{MR-00}). In addition, we charac…
This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation …
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