Narrow Results

Let P ∈ Γ ⊂ X be the germ of a Gorenstein three-fold singularity and Γ a smooth curve through it such that the general hyperplane section S of X containing Γ is DuVal of type E₆ or E₇. In this paper we obtain criteria for the existence of a terminal divisorial extremal neighborhood contracting an irreducible divisor E onto Γ and classify all such neighborhoods.

In this paper, we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in ${\mathbb{P}}_{ℂ}^2$. As a corollary we get that every discrete compact surface group in ${\mathrm{\text{PO}}}^{+}(2,1)$ admits a deformation in $\mathrm{\text{PSL}}(3,ℂ)$ with a nonempty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on ${\mathbb{P}}_{ℂ}^2$.

In this paper, we prove formulas that represent two-pointed Gromov–Witten invariant ${〈{{𝒪}}_{h^a}{{𝒪}}_{h^b}〉}_{0,d}$ of projective hypersurfaces with $d=1,2$ in terms of Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, the moduli spaces of stable maps from genus $0$ stable curves to projective space ${\mathbb{P}}^{N-1}$. Our formulas are based on representation of the intersection number $w{({{𝒪}}_{h^a}{{𝒪}}_{h^b})}_{0,d}$, which was introduced by Jinzenji, in terms of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$, the moduli space of quasi maps from ${\mathbb{P}}^1$ to ${\mathbb{P}}^{N-1}$ with two marked points. In order to prove our formulas, we use the results on Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, that were derived by Mustaţǎ and Mustaţǎ. We also present explicit toric data of ${\widetilde{Mp}}_{0,2}(N,d)$ and prove relations of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$.

In this paper, we present explicit toric construction of moduli space of quasi maps from ${\mathbb{P}}^1$ with two marked points to ${\mathbb{P}}^1\times {\mathbb{P}}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

We compute all relative dynamical degrees of equivariant dominant rational maps on toric varieties. We use the intersection theory on toric varieties using Minkowski weights. As a result, we see that the relative dynamical degrees can be reduced to the dynamical degrees on toric varieties. Hence these are all algebraic integers.

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.

In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss–Manin system.

We observe a new puzzling physical phenomenon in F-theory on the multisection geometry, wherein a model without a gauge group transitions to another model with a discrete ${\mathbb{Z}}_n$ gauge group via Higgsing. This phenomenon may suggest an unknown aspect of F-theory compactification on multisection geometry lacking a global section. A possible interpretation of this puzzling physical phenomenon is proposed in this note. We also propose a possible interpretation of another unnatural physical phenomenon observed for F-theory on four-section geometry, wherein a discrete ${\mathbb{Z}}_2$ gauge group transitions to a discrete ${\mathbb{Z}}_4$ gauge group via Higgsing as described in the previous literature.

In this paper, we briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi–Yau landscape — accumulated by the collaboration of physicists, mathematicians and computer scientists over the last four decades — as a starting-point and concrete playground, we review some recent progress in machine-learning applied to the sifting through of possible universes from compactification, as well as wider problems in geometrical engineering of quantum field theories. In parallel, we discuss the program in machine-learning mathematical structures and address the tantalizing question of how it helps doing mathematics, ranging from mathematical physics, to geometry, to representation theory, to combinatorics and to number theory.

In this paper, we analyze gauge groups in six-dimensional $N=1$ F-theory models. We construct elliptic Calabi–Yau 3-folds possessing various singularity types as double covers of “1/2 Calabi–Yau 3-folds,” a class of rational elliptic 3-folds, by applying the method discussed in a previous study to classify the singularity types of the 1/2 Calabi–Yau 3-folds. One to three U(1) factors are formed in six-dimensional F-theory on the constructed Calabi–Yau 3-folds. The singularity types of the constructed Calabi–Yau 3-folds corresponding to the non-Abelian gauge group factors in six-dimensional F-theory are deduced. The singularity types of the Calabi–Yau 3-folds constructed in this work consist of $A$- and $D$-type singularities.

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

In this paper, we examine the large-$g$ asymptotic Weil–Petersson volume formulas deduced in the previous literature. The volume formulas have application to computing the partition functions and the correlation functions in Jackiw–Teitelboim gravity. We utilize two approaches to assess the validity of the formulas. The first approach is to examine the asymptotic volume formulas from the perspective of the Witten conjecture. If the volume formulas are correct, the generating function of the intersection indices deduced from the asymptotic volume formulas would satisfy constraints that are analogous to the Virasoro constraints. We confirmed that the intersection indices computed from the large-$g$ asymptotic volume formulas satisfy variants of the string and dilaton equations. This implies that the generating function of the intersection indices deduced from the asymptotic volume formulas indeed satisfies constraints analogous to first two of the Virasoro constraints. As another approach, we also examined the asymptotic volume formulas by studying the behavior of the higher-order spectral form factors. Our analyses suggest that the large-$g$ asymptotic Weil–Petersson volume formulas yield plausible estimations.

Kobayashi conjecture says that every holomorphic map is constant for a very general hypersurface D⊂ℙⁿ of degree deg D≥2n+1. As a corollary of our main theorem, we show that f is constant if is contained in an algebraic curve.

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

Suppose that $X_A\subset {\mathbb{P}}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper, we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix $A$ in the codimension two case.

Given a bivariate system of polynomial equations with fixed support sets $A,B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the range $\{0,1,\dots ,|A|-|conv(A)\ominus B|-1\}$, where $A\ominus B$ is the set of all integral vectors that shift B to a subset of $A$. As an application, we classify all pairs $(A,B)$ such that the system supported at $(A,B)$ does not have a solution of multiplicity higher than $2$.

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_{\mathrm{\Sigma }}$. The algorithm lends its name from a construction, described by Cox, of $X_{\mathrm{\Sigma }}$ as a GIT quotient $X_{\mathrm{\Sigma }}=({ℂ}^k\setminus Z)//G$ of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ${ℂ}^k$ of $X_{\mathrm{\Sigma }}$ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of $X_{\mathrm{\Sigma }}$. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the $G$-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of $X_{\mathrm{\Sigma }}$. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.

We study multihomogeneous spaces corresponding to ${\mathbb{Z}}^n$-graded algebras over an algebraically closed field of characteristic $0$ and their relation with the description of $T$-varieties.

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the so called "gravitational theories with covariant and contravariant connection and metrics", it is shown that a wide variety of 3rd, 4th, 5th, 7th, 10th- degree algebraic equations exists in gravity theory. This is important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Since the obtained cubic algebraic equations are multivariable, the standard algebraic geometry approach for parametrization of two-dimensional cubic equations with the elliptic Weierstrass function cannot be applied. Nevertheless, for a previously considered cubic equation for reparametrization invariance of the gravitational Lagrangian and on the base of a newly introduced notion of "embedded sequence of cubic algebraic equations", it is demonstrated that in the multivariable case such a parametrization is also possible, but with complicated irrational and non-elliptic functions. After finding the solutions of a system of first-order nonlinear differential equations, these parametrization functions can be considered also as uniformization ones (depending only on the complex uniformization variable z) for the initial multivariable cubic equation.

For any line bundle written as a subtraction of two ample line bundles, Siu’s inequality gives a criterion on its bigness. We generalize this inequality to a relative case. The arithmetic meaning behind the inequality leads to its application on algebraic dynamic systems, which is the equidistribution theorem of generic and small net of subvarieties over a function field.