Boundary Divisors Research Articles

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

Divisors and curves on logarithmic mapping spaces

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Newton–Okounkov bodies and minimal models for cluster varieties

Let Y be a (partial) minimal model of a scheme V with a cluster structure (of type A, X or of a quotient of A or a fibre of X). Under natural assumptions, for every choice of seed we associate a Newton–Okounkov body to every divisor on Y supported on Y∖V and show that these Newton–Okounkov bodies are positive sets in the sense of Gross, Hacking, Keel and Kontsevich [31]. This construction essentially reverses the procedure in [31] that generalizes the polytope construction of a toric variety to the framework of cluster varieties.In a closely related setting, we consider cases where Y is a projective variety whose universal torsor UTY is a partial minimal model of a scheme with a cluster structure of type A. If the theta functions parametrized by the integral points of the associated superpotential cone form a basis of the ring of algebraic functions on UTY and the action of the torus TPic(Y)⁎ on UTY is compatible with the cluster structure, then for every choice of seed we associate a Newton–Okounkov body to every line bundle on Y. We prove that any such Newton–Okounkov body is a positive set and that Y is a minimal model of a quotient of a cluster A-variety by the action of a torus.Our constructions lead to the notion of the intrinsic Newton–Okounkov body associated to a boundary divisor in a partial minimal model of a scheme with a cluster structure. This notion is intrinsic as it relies only on the geometric input, making no reference to the auxiliary data of a valuation or a choice of seed. The intrinsic Newton–Okounkov body lives in a real tropical space rather than a real vector space. A choice of seed gives an identification of this tropical space with a vector space, and in turn of the intrinsic Newton–Okounkov body with a usual Newton–Okounkov body associated to the choice of seed. In particular, the Newton–Okounkov bodies associated to seeds are related to each other by tropicalized cluster transformations providing a wide class of examples of Newton-Okounkov bodies exhibiting a wall-crossing phenomenon in the sense of Escobar–Harada [18].This approach includes the partial flag varieties that arise as minimal models of cluster varieties (for example full flag varieties and Grassmannians). For the case of Grassmannians, our approach recovers, up to interesting unimodular equivalences, the Newton–Okounkov bodies constructed by Rietsch–Williams in [56].

Line bundles on the moduli space of Lie algebroid connections over a curve

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid (L) connections on a compact Riemann surface X of genus g≥3. A smooth compactification of the moduli space of L-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of L-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.

Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability

Abstract In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$ , these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$ , we prove that they are different by exhibiting some walls.

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

AbstractSmooth minimal surfaces of general type with , , and constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space of their canonical models admits a modular compactification via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of and the Hodge theory of the degenerate surfaces that the eight divisors parameterize.

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

Some examples of log Fano structures on blow-ups along subvarieties in products of two projective spaces

We give a series of examples of log Fano manifolds in any dimension greater than or equal to three by using successive blow-ups along subvarieties in products of two projective spaces. More precisely, we first blow-up a product of two projective spaces along a smooth hypersurface contained in a fiber of one of the two projections and then along the strict transform of a fiber (intersecting the center of the first blow-up) of the other projection. By computing the nef cone and giving an explicit boundary divisor, we show that the resulting variety is always a log Fano manifold. Moreover, the effective cone and the extremal contractions are described. Note that this variety depends on three integral parameters: the dimensions of the two projective spaces and the degree of the hypersurface, which is the center of the first blow-up. Hence, we obtain a series of examples of log Fano manifolds. We also determine which ones are weak Fano.

Classification of noncommutative monoid structures on normal affine surfaces

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.

Gromov–Witten theory and invariants of matroids

We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

Nearby cycles and semipositivity in positive characteristic

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

Boundary complexes of moduli spaces of curves in higher genus

Given a collection of boundary divisors in the moduli space M ¯ 0 , n \overline {\mathcal {M}}_{0,n} of stable genus-zero n n -pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs ( g , n ) (g,n) for which the analogous statement holds in M ¯ g , n \overline {\mathcal {M}}_{g,n} .

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.

On finiteness of log canonical models

Let [Formula: see text] be klt pairs with [Formula: see text] a convex set of divisors. Assuming that the relative Kodaira dimensions of such pairs are non-negative, then there are only finitely many log canonical models when the boundary divisors vary in a rational polytope in [Formula: see text]. As a consequence, we show the existence of the log canonical model for a klt pair [Formula: see text] with real coefficients.

Mirror symmetry for very affine hypersurfaces

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory along the skeleton of the hypersurface. Using Mikhalkin-Viro patchworking, we identify the skeleton of the hypersurface with the boundary of the Fang-Liu-Treumann-Zaslow skeleton. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

AbstractFor the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

On some modular contractions of the moduli space of stable pointed curves

The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves. These new moduli spaces, which are modular compactifications of the moduli space of smooth pointed curves, are related with the minimal model program for the moduli space of stable pointed curves and have been introduced in a previous work of the authors. We interpret them as log canonical models of adjoints divisors and we then describe the Shokurov decomposition of a region of boundary divisors on the moduli space of stable pointed curves.

Symmetries of tropical moduli spaces of curves

We compute the automorphism group $\mathrm{Aut}(\Delta_{g, n})$ for all $g, n \geq 0$ such that $3g - 3 + n > 0$, where $\Delta_{g, n} \subset M_{g, n}^\mathrm{trop}$ is the moduli space of stable $n$-marked tropical curves of genus $g$ and volume one. In particular, we show that $\mathrm{Aut}(\Delta_{g})$ is trivial for $g \geq 2$, while $\mathrm{Aut}(\Delta_{g, n}) \cong S_n$ when $n \geq 1$ and $(g, n) \neq (0, 4), (1, 2)$. The space $\Delta_{g, n}$ is a symmetric $\Delta$-complex in the sense of Chan, Galatius, and Payne, and is identified with the dual intersection complex of the boundary divisor in the Deligne-Mumford-Knudsen moduli space $\overline{\mathcal{M}}_{g, n}$ of stable curves. After the work of Masseranti, who has shown that $\mathrm{Aut}(\overline{\mathcal{M}}_g)$ is trivial for $g \geq 2$ while $\mathrm{Aut}(\overline{\mathcal{M}}_{g, n}) \cong S_n$ when $n \geq 1$ and $2g - 2 + n \geq 3$, our result implies that the tropical moduli space $\Delta_{g, n}$ faithfully reflects the symmetries of the algebraic moduli space for general $g$ and $n$.

Boundary contributions to three loop superstring amplitudes

In type II superstring theory, the vacuum amplitude at a given loop order g can receive contributions from the boundary of the compactified, genus g supermoduli space of curves M̄g. These contributions capture the long distance or infrared behavior of the amplitude. The boundary parameterizes degenerations of genus g super Riemann surfaces. A holomorphic projection of the supermoduli space onto its reduced space would then provide a way to integrate the holomorphic, superstring measure and thereby give the superstring vacuum amplitude at g-loop order. However, such a projection does not generally exist over the bulk of the supermoduli spaces in higher genera. Nevertheless, certain boundary divisors in ∂M̄g may holomorphically map onto a bosonic space upon composition with universal morphisms, thereby enabling an integration of the holomorphic, superstring measure here. Making use of ansatz factorizations of the superstring measure near the boundary, our analysis shows that the boundary contributions to the three loop vacuum amplitude will vanish in closed oriented type II superstring theory with unbroken spacetime supersymmetry.