This paper shows that the multiplicity of the base points locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base points multiplicity, and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.
We study the singular locus on the algebraic surface $§_n$ of genus 2 curves with a $(n, n)$-split Jacobian. Such surface was computed by Shaska in \cite{deg3} for $n=3$, and Shaska at al. in \cite{deg5} for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface $§_3$ discovered in \cite{deg3} to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We furt…
We prove that the set of singular configurations of a general Gough Stewart platform has a rational parametrization. We introduce a reciprocal twist mapping which, for a general orientation of the platform, realizes the cubic surface of singularities as the blowing up of a quadric surface in five points.
Given a unirational parameterization of a surface, we present a general algorithm to determine a birational parameterization without using parameterization algorithms. Additionally, if the surface is assumed to have a birational parametrization with empty base locus, and the input parametrization is transversal, the degree of the solution is determined in advance and the dimension of the space of solutions is reduced. As a consequence, for these cases, we present a second faster algorithm.
Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a parametrization over the reals can be given by cubic polynomials. In this short note we provide a simple geometric method to obtain such parametrization based in an algorithm by I. Polo-Blanco and J. Top \cite{Po-Top}.
Let $S$ be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps $f,g,h:\mathbb{A}^2 --\to S\subset \mathbb{P}^n$ such that the union of the three images covers $S$. As a consequence, we present a second algorithm that generates two rational maps $f,\tilde{g}:\mathbb{A}^2 --\to S$, such that the union of their images covers the affine surface $S\cap \mathbb{A}^n$. In the affine case, the number of rational maps involved in the cover is in general…
In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in Macaulay2. We showed in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Botbol-09] we generalized this to the case where the base locus is not necessarily a local complete intersection. The key point consists in exploiting the sparse structu…
We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.
We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.
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 birational transformations P^n--->S \subseteq P^N defined by linear systems of quadrics whose base locus is smooth and irreducible of dimension \leq3 and whose image S is sufficiently regular.
We present a new class of examples of base points for the generalized theta divisor on the moduli space of semistable vector bundles of trivial determinant on a compact Riemann surface and we prove that for sufficiently large rank the base locus is positive dimensional.
We produce a lower bound for the dimension of the base locus of the generalized theta divisor on the moduli space SU_C(r) of semistable vector bundles of rank r and trivial determinant on a smooth curve C of genus g > 1.
Let $L$ be a nef and big line bundle on a scheme $X$. It is well known that if $X$ is a projective over a field then the augmented base locus and the exceptional base locus agree. This result is extended to projective schemes over arbitrary excellent Noetherian bases, assuming the result holds in characteristic zero. In particular the result holds if the base is a mixed characteristic Dedekind domain or if $L$ is semiample in characteristic $0$.
In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree two and three stable maps to Grassmannians. This gives new examples of the decomposition for varieties with Picard rank three. We also discuss the birational models that correspond to the chambers in the decomposition.
We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers.
Consider a (non-empty) linear system of surfaces of degree d in P^3 through at most 8 multiple points in general position and let L denote the corresponding complete linear system on the blowing-up X of P^3 along those general points. Then we determine the base locus of such linear systems L on X.
We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results.
This paper shows that on the moduli space of semi-stable vector bundles of fixed rank and determinant (of any degree) on a smooth curve of genus at least two, the base locus of the generalized theta divisor is large provided the rank is sufficiently large. It also shows that the base locus is large for positive multiples of the theta divisor. This work extends results already known for the case where the determinant is of degree zero.
Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that the restriction of L to V is not big. For a smooth variety in characteristic zero, this was proved by Nakamaye using vanishing theorems.
Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the determinant line bundle $\cL$ over $\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\cB_{g+2}$ over any curve $C$, and base points in $\cB_4$ over any hyperelliptic curve.
In this paper, we first introduce geometric operations for linear categories, and as a consequence generalize Orlov's blow up formula [O04] to possibly singular local complete intersection centres. Second, we introduce refined blowing up of linear category along base--locus, and show that this operation is dual to taking linear section. Finally, as an application we produce examples of Calabi--Yau manifolds which admits Calabi--Yau categories fibrations over projective spaces.
Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$, or is Zariski-dense in $S$.
In this paper, we present an algorithm for reparametrizing birational surface parametrizations into birational polynomial surface parametrizations without base points, if they exist. For this purpose, we impose a transversality condition to the base points of the input parametrization.
Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate Néron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.
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 …
We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.
We exhibit a pseudoeffective R-divisor D_λon the blow-up of P^3 at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus B_-(D_λ) = \bigcup_{A ample}} B(D_λ+A) is not closed and that D_λdoes not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an R-divisor on the family of blow-ups of P^2 at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.
We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result does not need $Y$ to exist as a variety, i.e. it may be "noncommutative". We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a categorical resolution of singularities of $Y_L$…
The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our …
The purpose of this paper is to incorporate minimal free resolutions into the study of the birational geometry of the moduli space of coherent sheaves on the plane with character $ξ$, denoted by $M(ξ)$. We show that it is possible to recover the relevant Bridgeland destabilizing object from the minimal free resolution in order to compute the effective cone $\mathrm{Eff}(M(ξ))$ and conjecture a full relationship between Bridgeland destabilizing objects and minimal free resolutions. Moreover, we also prove that minimal free resolutions, paired with interpolation for vector bundles, yield the mov…
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…
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 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 interpret geometrically the torsion of the symmetric algebra of the ideal sheaf of a zero-dimensional scheme Z defined by $n+1$ equations in an $n$-dimensional variety. This leads us to generalise a formula of A.Dimca and S.Papadima in positive characteristic for a rational transformation with finite base locus. Among other applications, we construct an explicit example of a homaloidal curve of degree $5$ in characteristic $3$, answering negatively a question of A.V.Dória, S.H.Hassanzadeh and A.Simis.
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.
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.
We provide a locally free resolution of the projectivized symmetric algebra of the ideal sheaf of a zero-dimensional scheme defined by n + 1 equations in an n-dimensional variety. The resolution is given in terms of the resolution of the ideal itself and of the Eagon-Northcott complex of the Koszul hull.
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.
We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.
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…
In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in …
We show that the implicit equation of a surface in 3-dimensional projective space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a given positive integer d, can be represented and computed from the linear syzygies of its parametrization if the base points are isolated and form locally a complete intersection.
