The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map $ε$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($ε$ sends the normal vector $O-x$ to its endpoint $O$), and in this paper we address two general problems :
1) Characterize the "degenerate" case where the focal locus is not a hypersurface
2) Calculate, in the case where $Σ_X$ is a hypersurface, its degree (with multiplicity)
In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve C on a general surface in P^3 of degree d>4 has geometric genus g > 1 + deg(C)(d - 5)/2. Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in P^3 and on a general projectively Cohen-Macaulay surface in P^4.
We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties.
We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure.
We extend the notion of absolute subsets of Betti moduli spaces of smooth algebraic varieties to the case of normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over a normal variety are a finite union of translated subtori. We show that the same holds for jump loci twisted by a unitary local system in the case where the underlying variety $X$ is projective with $H^1(X,\mathbb{Q})$ pure of weight one. Lastly, we study the interaction of these loci with Hodge theoretic data naturally associated to the representation variety of fundamental groups of smooth p…
Linear projections from P^k to P^h appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of many enough correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in P^k containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical project…
Given the Prym variety of an étale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the covering. In this paper we describe the singular locus and the irreducible components of $V^2$.
For the toric variety X associated to the Bruhat poset of Schubert varieties in a minuscule G/P, we describe the singular locus in terms of the faces of the associated polyhedral cone. We further show that the singular locus is pure of codimension 3 in X.
For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.
We prove that the non-nef locus and the restricted base locus of a pseudoeffective divisor coincide on KLT pairs. We also extend to KLT pairs F. Russo's characterization of nef and abundant divisors by means of asymptotic multiplier ideals.
We prove that the cohomology jump loci in the space of rank one local systems over a smooth quasi-projective variety are finite unions of torsion translates of subtori. The main ingredients are a recent result of Dimca-Papadima, some techniques introduced by Simpson, together with properties of the moduli space of logarithmic connections constructed by Nitsure and Simpson.
In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after suitable twists by powers of $L$. Moreover we show that similar results hold for subvarieties of a complex abelian variety.
We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed-invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the $G_2$-variety, including its quantum version. We also consider the Spin$_7$-variety, which deserves a different treatment.
Let $X$ be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of $V^0(ω_X)$ generate $\Pic^0(X)$. We then study the pluricanonical maps of $X$. We show that $|4K_X|$ induces a birational map.
In this paper we construct Shimura subvarieties of dimension bigger than one of the moduli space of polarised abelian varieties of a given dimension, which are generically contained in the Pym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely to use families of Galois covers of the projective line. The case of abelian covers is treated in detail, since in this case it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield…
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.
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 define the notion of isosingular loci of algebraic varieties, following the analytic case first studied by Ephraim. In particular, we give a partial extension of his main result in arbitrary characteristic and a full extension assuming characteristic $0$. One of the main obstructions in the positive characteristic case is the non-separability of the orbit map associated to the contact group, as first observed by Greuel and Pham for isolated singularities.
We show that the dual of the cone of divisors on a complete Q-factorial toric variety X whose stable base loci have dimension less than k is generated by curves on small modifications that move in families sweeping out the birational transforms of k-dimensional subvarieties of X. We give an example showing that it does not suffice to consider curves on X itself.
We describe the non-Gorenstein loci of normal toric varieties. In the case of Hibi rings a combinatorial description is provided in terms of the underlying partially ordered set. As a non-toric application we compute the dimensions of the non-Gorenstein loci of the first secant variety of Segre varieties.
We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one. As a further application, we establish a nonabelian Hodge correspondence on smooth loci of klt varieties.
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.
We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized …
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 $E$.
We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 …
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
In this paper, improving on results of Abramovich- Harris and Debarre- Fahlaoui we give the full classification of curves C of genus g such that a Brill--Noether locus W^ s_d(C), strictly contained in the jacobian J(C) of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety A contained in J(C) and such that dim(Z)=d-dim(A)-2s, i.e., the maximum possible for such a Z.
For the toric variety X associated to the Bruhat poset of Schubert varieties in the Grassmannian, we describe the singular locus in terms of the faces of the associated polyhedral cone. We also determine the tangent cones at the maximal singularities of X. These turn out to be again toric varieties. In the case of X being associated to the Bruhat poset of Schubert varieties in the Grassmannian of 2-planes in a n-dimensional vector space (over the base field), we also prove a certain product formula relating the multiplicities at certain singular points.
We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution and its fixed points. These results allow us to give a new proof of the statements about the dimensions, for small genus, of the loci of ppav with theta divisor containing two-torsion points of multiplicity three. We also prove a result about the closure of this set. Our computations used in our work arXiv:1103.1857 to compute the class of the closure of the …
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.
Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) I give a uniform description (that is, independent of type) of the irreducible components of the singular locus of a Schubert variety Y in X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties by an non-negative integer A and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if A=0; (2) if G of Type ADE, then the singular locus occurs in codimension at least t…
In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual varieties carries over to a connection between higher order polar loci and higher order dual varieties. We generalize the duality between the degrees of polar classes of a variety and those of its dual variety to varieties that are reflexive to a higher order. In particular, the degree of the top (highest codimension) polar class of order k is equal to the degree…
In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.
Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$ as a generator of $Pic(X)$ and the genus $g$ of $C$ is sufficiently high, we show the Brill-Noether locus $BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperkähler manifold of dimension $2g -2r \lfloor \frac{g}{r}\rfloor$, isomorphic to a moduli space of stable vector bundles on $X$.
We extend to normal projective varieties defined over an arbitrary algebraically closed field a result of Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa characterizing the augmented base locus (aka non-ample locus) of a line bundle on a smooth projective complex variety as the union of subvarieties on which the restricted volume vanishes. We also give a proof of the folklore fact that the complement of the augmented base locus is the largest open subset on which the Kodaira map defined by large and divisible multiples of the line bundle is an isomorphism.
We give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X_w for any element w in S_n. Our description of the irreducible components is computationally more efficient (O(n^6)) than the previously best known algorithms. This result proves a conjecture of Lakshmibai and Sandhya regarding this singular locus. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.
We study entry loci of varieties and their irreducibility from the perspective of $X$-ranks with respect to a projective variety $X$. These loci are the closures of the points that appear in an $X$-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in $\mathbb P^4$ using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with…
Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We use this result to solve the constant rank problem for symmetric matrices, proving that the maximal dimension of a linear subspace of the space of $m\times m$ symmetric matrices such that each nonzero element has even rank $r \ge 2$ is $m-r+1$. We explain how this result relates t…
Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and Demmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-based representation that characterizes the non-negativity of $f$ on $V$. This representation allows us to build up exact semidefinite relaxations for polynomial optimization problems whose optimal solutions are possibly singularities of the constraint sets.
We study second order focal loci of nondegenerate plane congruences in P4(C) with degenerate focal conic. We show the projective generation of such congruences when the second order focal locus fills a component of the focal conic, improving a result of Corrado Segre
We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkaehler quiver varieties, as fixed loci of such actions.
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local version of the theorem. This yields the "Hodge decomposition" of the set of unitary local systems with a non-vanishing cohomology extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider a twisted version of the characteristic varieties generalizing the twisted Alexander polynomi…
We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.
Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their "group-theoretic models" -- generalized affine Deligne-Lusztig varieties -- in cases where they have a particularly nice description. Continuing the work of [GH] and [GHN] we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new and open cases from the point of view of Shimura varieties/Rapoport-Zink spaces.
In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \subset ¶^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \geq 2$, $q \in ¶^r$ a closed point outside of $X$, and $X_q := π_q (X) \subset ¶^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $Σ_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on $X$ spanning secant lines passing through $q$, is either empty or a quadric in a subspace of $¶^r$.…
The multiple root loci among univariate polynomials of degree $n$ are indexed by partitions of $n$. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our emphasis lies on equations and parametrizations that are useful for Euclidean distance optimization. We compute the ED degrees for hooks. Among the dual hypersurfaces are those that demarcate the set of binary forms whose real rank equals the generic complex rank.
We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: the characteristic varieties, or, the support loci for homology with coefficients in rank 1 local systems, and the resonance varieties, or, the support loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X. The ge…
In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $ |H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\…
We give a complete (global) characterization of complex perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell's work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser's results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.
In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-s…
