The aim of this paper is to study Weil divisors on a singular rational normal scroll X. In particular the author describes explicitly the group of divisorial sheaves associated to Weil divisors on X, via the direct image of the Picard group of the canonical resolution of X, which is very well known.
The analysis naturally splits in two cases, depending on the codimension of the vertex of X (>2 or =2). When the codimension of the vertex is >2, then the two groups are isomorphic and there is a natural intersection form in X inherited from the one in the canonical resolution. When the codimension of the vertex is 2 this is no longer true and to overcome this problem the author introduces the concept of integral total transform of a divisor in the resolution of X. In this case the author defines a (non-linear) intersection number of two divisors, that represents the degree of their scheme-theoretic intersection if they are effective with no common components. Lastly there are some examples and applications.
The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.
We define and study positivity (nefness, amplitude, bigness and pseudo-effectiveness) for Weil divisors on normal projective varieties. We prove various characterizations, vanishing and non-vanishing theorems for cohomology, global generation statements, and a result related to log Fano.
We study the asymptotics of the Weil-Petersson volumes of the moduli spaces of compact Riemann surfaces of genus $g$ with $n$ punctures, for fixed $n$ as $g \to \infty$.
Mumford defined a rational pullback for Weil divisors on normal surfaces, which is linear, respects effectivity, and satisfies the projection formula. In higher dimensions, the existence of small resolutions of singularities precludes such general results. We single out a higher-dimensional situation that resembles the surface case and show for it that a rational pullback for Weil divisors exists, which is also linear, respects effectivity, and satisfies the projection formula.
We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution $\tilde{X}\to X $ and the corresponding canonical divisor. We provide combinatorial formulae for the delta--invariant $δ(X,0)$ and for the cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case $\d…
Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's Geometric Invariant Theory. We obtain several new Hilbert-Mumford type theorems, and we extend a projectivity criterion of Bialynicki-Birula and Swiecicka for varieties with semisimple group action from the smooth to the singular case.
In this paper, we study the minimal free resolution of non-ACM divisors $X$ of a smooth rational normal surface scroll $S=S(a_1 ,a_2 ) \subset \mathbb{P}^r$. Our main result shows that for $a_2 \geq 2a_1 -1$, there exists a nice decomposition of the Betti table of $X$ as a sum of much simpler Betti tables. As a by-product of our results, we obtain a complete description of the graded Betti numbers of $X$ for the cases where $S=S(1,r-2)$ for some $r \geq 3$ and $S=S(2,r-3)$ for some $r \geq 6$.
We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can …
Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.
Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over …
A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.
We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on homogeneous spectra defined by Weil divisors on surfaces.
We introduce a notion of volume of a normal isolated singularity that generalizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion…
Given a Weil non-integral divisor $D$, it is natural to associate it the line bundle of its integral part $\mathcal{O}_X([D])$. In this work we study which of the classical characterizations of ample and big divisors can be extended to non-integral divisors via the corresponding line bundles.
In this paper, we investigate the explicit birational geometry for projective $ε$-lc varieties polarised by nef and big Weil divisors. We show that if $X$ is a projective $ε$-lc variety, $H$ is a nef and big Weil divisor with $\dim\overline{φ_{H}(X)}\geq n-1$ and $L$ is an effective Weil divisor such that $|L-K_X|\neq \emptyset$ or $L-K_X$ is nef, then we can find an explicit lower bound of $\text{vol}(H)$ and prove that $|L+m^\prime H|$ is birational for $m^\prime\geq m$, where $m$ is an explicit number which depends only on $n$ and $ε$. This result can be applied to polarised Calabi-Yau vari…
In this paper we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if $X$ is a projective variety of dimension $d$ with $ε$-lc singularities for $ε>0$, and if $N$ is a nef and big Weil divisor on $X$ such that $N-K_X$ is pseudo-effective, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d,ε$. This is key to proving various other results. For example, it implies that if $N$ is a big Weil divisor (not necessaril…
Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.
In this paper we investigate the divisor $\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus…
For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.
The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studi…
We consider normal projective n-dimensional varieties X whose anticanonical divisor class -K is ample and where every Weil divisor is a rational multiple of K. The index i is the largest integer such that K/i exists as a Weil divisor. We show (i) if X has log-terminal singularities, and in addition 1-forms on the smooth part of X are holomorphic on a resolution, then (-K)^n =< (max(in,n+1))^n; (ii) if the tangent sheaf of X is semistable, then (-K)^n =<(2n)^n. The proof is based on some elementary but possibly surprising slope estimates on sheaves of differential operators on plurianticanonica…
Let G be a finite subgroup of GL_n(C). A study is made of the ways in which resolutions of the quotient space C^n / G can parametrise G-constellations, that is, G-regular finite length sheaves. These generalise G-clusters, which are used in the McKay correspondence to construct resolutions of orbifold singularities. A complete classification theorem is achieved, in which all the natural families of G-constellations are shown to correspond to certain finite sets of G-Weil divisors, which are a special sort of rational Weil divisor, introduced in this paper. Moreover, it is shown that the number…
In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.
This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2-package which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on a dedicated webpage. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all…
We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor due to Hashimoto.
Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
Let $X$ be a general divisor in $|3M+nL|$ on the rational scroll $\mathrm{Proj}(\oplus_{i=1}^{4}\mathcal{O}_{\mathbb{P}^{1}}(d_{i}))$, where $d_{i}$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural projection to $\mathbb{P}^{1}$, and $d_{1}\geqslant...\geqslant d_{4}=0$. We prove that $X$ is rational $\iff$ $d_{1}=0$ and $n=1$.
In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrictions of line bundles on the variety. As a consequence, we describe the group of sections of the Hitchin fibration for moduli spaces of Higgs bundles on curves.
In this paper, we study properties of Weil height functions associated with numerically trivial divisors. It helps us to define the fractional limit of $h_E$ with respect to $h_D$ on $U$, with $D$ ample: \[ \Flim_D(E,U) := \liminf_{\substack{P \in U h_D(P) \rightarrow \infty}}\dfrac{h_E(P)}{h_D(P)}. \] The value of $\Flim_D(E,U)$ contains numerical information about a divisor $E$, enough to determine whether $E$ is ample, numerically effective or pseudo-effective.
We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geq 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.
This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not Q-factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier divisors are generated by "topological traces " of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces wi…
Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then reduced to the case of $\mathop{Ext}\nolimits^1$ via a quive…
Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, Δ)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $Δ$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.
We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls, and twists thereof, and determine the action of the discriminant or big diagonal divisor on this group by intersection. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry. The latest version inc…
The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more gen…
We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I$_2$+I$_4$ collision of singularities. The resulting gauge theory has a semi-simple Lie algebra $\mathfrak{su}(2)\oplus \mathfrak{sp}(4)$ or $\mathfrak{su}(2)\oplus \mathfrak{su}(4)$. We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometric…
Let X be a minuscule Schubert variety and $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The irreducible components are described in the following way : the class $α$ can be seen as an element of $Pic(X)^*$ the dual of the Picard group. Because any Weil-divisor need not to be a Cartier-divisor, there is (only) a surjective map $s:A^1(X)^*\to Pic(X)^*$ from the dual of the group of codimension 1 cycles to the dual of the Picard group. The irreducible components are given by the effective ele…
A description of complete normal varieties with lower dimensional torus action has been given by Altmann, Hausen, and Suess, generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T-varieties to define new evaluation codes called T-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory…
We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tate's conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.
Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumfor…
For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that $H_{\mathrm{ar}}^0(X, \mathcal O_X (D))$ is discrete, $H_{\mathrm{ar}}^1(X, \mathcal O_X (D))$ is finite, and $H_{\mathrm{ar}}^2(X, \mathcal O_X (D))$ is compact. Moreover, we prove that all possible summations of canonical subspaces $\mathbb A_{X,i}^{\mathrm{ar}}(D),$ $\mathbb A_{X, kl}^{\mathrm{ar}}(D)$ $(i,k,l=0,1,2)$ are closed in $\mathbb A_{X,012}^{\m…
In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms $Ψ(F)$ with product expansions on bounded domains $D$ associated to rational quadratic spaces $V$ of signature (n,2). The input $F$ for his construction is a vector valued modular form of weight $1-n/2$ for $SL_2(Z)$ which is allowed to have a pole at the cusp and whose non-positive Fourier coefficients are integers $c_μ(-m)$, $m\ge0$. For example, the divisor of $Ψ(F)$ is the sum over $m>0$ and the coset parameter $μ$ of $c_μ(-m) Z_μ(m)$ for certain rational quadratic divisors $Z_μ(m)$ on the arith…
Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $α$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $α$ from a rational curve to X. The situation depends on the line bundle L : if the projectivised tangent space to the vertex contains lines (i.e. if G/Q contains lines in P) then the irreducible components are described as in our paper math.AG/0407123 by the…
Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X,D) is said to be semi-simple normal crossings (semi-snc) at a point a of X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f: X' -> X such that the transformed pair (X',D') is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal…
We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell-Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z/2Z for SO(4). Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating func…
