Suppose D is an effective divisor on a smooth projective algebraic variety X. For each point x of X we associate a numberical invariant called the moving Seshadri constant of D at x which is a numerical measure of positivity of the divisor D at x. We determine the base locus of a large multiple of D by studying these moving Seshadri constants-- the drawback to this method is that these moving Seshadri constants are extremely difficult to compute.
We determine the fixed locus of the anticanonical complete linear system of a given anticanonical rational surface. The case of a geometrically ruled rational surface is fully studied, e.g., the monoid of numerically effective divisor classes of such surface is explicitly determined and is minimally generated by two elements. On the other hand, as a consequence in the particular case where $X$ is a smooth rational surface with $K_{X}^{2}>0$, the following expected result holds: every fixed prime divisor of the complete linear system $|-K_X|$ is a $(-n)$-curve, for some integer $n\geq 1$.
We give equivalent descriptions for the augmented and diminished base loci of vector bundles in characteristic zero. We show that these base loci behave well under pullback, tensor product, and direct sum. Pathological behavior is observed on some nonsplit exact sequences.
We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.
The purpose of this paper is to investigate the behaviour of certain asymptotic invariants of line bundles on projective surfaces. In particular, we describe the volume of line bundles and their destabilizing numbers.
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.
Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this note I refine the work in arXiv:math/0406288v1 and answer this question under a divisibility assumption. To do this I translate the algebraic statement into a geometric one concerning the base loci of linear systems of $P^n$ with assigned singularities.
The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the l…
We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.
We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtai…
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 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.
In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null locus. We also show a generalized Fujita approximation for the arithmetic restricted volumes.
An enhancement of the dynamic geometry system GeoGebra for the automatic symbolic computation of algebraic loci and envelopes is presented. Given a GeoGebra construction, the prototype, after rewriting the construction as a polynomial system in terms of variables and parameters, uses an implementation of the recent GröbnerCover algorithm to obtain the algebraic description of the sought locus/envelope as a locally closed set. The prototype shows the applicability of these techniques in general purpose dynamic geometry systems.
An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the siz…
In this paper, we describe the osculating cone to Brill-Noether loci $W^{0}_{d}(C)$ at smooth isolated points of $W^{1}_{d}(C)$ for a smooth canonically embedded curve $C$ of even genus $g=2(d-1)$. In particular, we show that the canonical curve $C$ is a component of the osculating cone. The proof is based on techniques introduced by George Kempf in 1986.
We compute the $δ$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic algebraic spatial curve projections). Our approach is based on some new results on zero loci of Schur polynomials, on transversality properties of maps defined by sparse polynomials, and on a new refinement of the notion of tropicalization of a curve (ultratropicalization), which may be of independent interest.
Let $(X,B)$ be a projective log canonical pair such that $B$ is a $\Q$-divisor, and that there is a surjective morphism $f\colon X\to Z$ onto a normal variety $Z$ satisfying: $K_X+B\sim_\Q f^*M$ for some $\Q$-divisor $M$, and the augmented base locus ${\bf{B_+}}(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.
We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers. Our methods indicate a link between $p$-adic measure and the minimal model theory.
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 …
The bad locus and the rude locus of an ample and base point free linear system on a smooth complex projective variety are introduced and studied. The bad locus is defined as the set of points that force divisors through them to be reducible. The rude locus is defined as the set of points such that divisors that are singular at them are forced to be reducible. The existence of a nonmempty bad locus is shown to be exclusively a two dimensional phenomenon. Polarized surfaces of small degree, or whose degree is the square of a prime, with nonempty bad loci are completely classified. Several explic…
We study the cone of Moriwaki divisors on \bar{M}_g by means of augmented base loci. Using a result of Moriwaki, we prove that an R-divisor D satisfies the strict Moriwaki inequalities if and only if the augmented base locus of D is contained in the boundary of \bar{M}_g. Then we draw some interesting consequences on the Zariski decomposition of divisors on \bar{M}_g, on the minimal model program of \bar{M}_g and on the log canonical models \bar{M}_g(α).
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.
For an algebraic K3 surface with complex multiplication (CM), algebraic fibres of the associated twistor space away from the equator are again of CM type. In this paper, we show that algebraic fibres corresponding to points at the same altitude of the twistor base $S^2\simeq \mathbb{P}^1_\mathbb{C}$ share the same CM endomorphism field. Moreover, we determine all the admissible Picard numbers of the twistor fibres.
Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.
We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and …
The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions a…
We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in de…
In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $ϕ: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana--Cao--Matsumura's equality on Hacon--$\rm{M^c}$…
Let $X$ be a normal projective variety defined over an algebraically closed field and let $Z$ be a subvariety. Let $D$ be an $\mathbb R$-Cartier $\mathbb R$-divisor on $X$. Given an expression $(\ast) \ D \sim_{\mathbb R} t_1 H_1 + \ldots + t_s H_s$ with $t_i \in \mathbb R$ and $H_i$ very ample, we define the $(\ast)$-restricted volume of $D$ to $Z$ and we show that it coincides with the usual restricted volume when $Z \not\subseteq \mathbf{B}_+(D)$. Then, using some recent results of Birkar (arXiv:1312.0239), we generalize to $\mathbb R$-divisors the two main results of arXiv:1305.4284 by Bou…
The bad locus of a base-point free linear system L on a normal complex projective variety X is defined as the subset B(L) of X of points that are not contained in any irreducible and reduced member of L. In this paper we provide a geometric description of such locus in terms of the morphism defined by L. In particular, assume that the dimension of X is at least 2, and that L is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(L) is empty unless X is a surface. Then we prove that, when the latter occurs, B(L) is not empty if and only if …
We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.
In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.
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.
The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(…
Working with a singular variety X, one is often interested in comparing the set of etale covers of X with that of its smooth locus X_reg. More precisely, one may ask: What are the obstructions to extend finite etale covers of X_reg to all of X? How do the etale fundamental groups of X and of its smooth locus compare? For projective varieties with Kawamata log terminal (klt) singularities we answer these questions in part. The main result of this paper asserts that there are no infinite towers of finite morphisms over a klt base variety where all morphisms are etale in codimension one, but bran…
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.
We define two tau functions, $τ$ and $\hatτ$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $τ$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hatτ$ is used to compute the divisor of canonical 1-forms with multiple zeros.
We compare and contrast results of E. Davis, of A. Bigatti, A.V. Geramita and the author, and of J. Ahn and the author. The underlying idea is that certain numerical conditions on the Hilbert function of a finite set of points in projective space force geometric consequences on the base loci of the linear systems of hypersurfaces containing the points. When the points have uniform position, this tends to force better behavior for these base loci. However, unexpected behavior is still possible, and we give examples.
We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the augmented base locus generalizing a result of Keel. We also study nef divisors with positive top intersection number, and discuss some problems related to augmented base loci of log divisors.
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.
The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently infinite constructions. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel'fand-Gel'fand, we give an alternate description in finite terms. Using this description we can characterize the generic complex, over the variety of finite free complexes of a given shape, as the direct image of an easily-described vector bundle. We can …
We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if t…
