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Sz-Sheng Wang

Publications and source records attributed to Sz-Sheng Wang.

11 recordsLinked to original sources
arXiv2026

Fiber products under toric flops and flips

Let $Σ$ and $Σ'$ be two refinements of a fan $Σ_0$ and $f \colon X_Σ \dashrightarrow X_{Σ'}$ be the birational map induced by $X_Σ \rightarrow X_{Σ_0} \leftarrow X_{Σ'}$. We show that the graph closure $\overlineΓ_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality. In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_Σ\mathop{\times}\limits_{X_{Σ_0}}X_{Σ'}$ is in general not toric, though it is still irreducible and $X_{\rm red} = \overlineΓ_f$. A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_Σ$ has at most terminal singularities. In this case, we further conclude that $X$ is normal.

math.AG↗
arXiv2025

Quantum Extremal Transitions and Special L-values

A threefold extremal transition $Y \searrow X$ consists of a crepant extremal contraction $ϕ\colon Y \to \bar Y$ with curve class $\ell \in \operatorname{NE}(Y)$, followed by a smoothing $\bar Y\rightsquigarrow X$. We consider the Type II case that $ϕ$ contracts a divisor $E$ to a point and prove that the quantum cohomology $QH(X)$ is obtained from $QH(Y)$ via analytic continuation, regularization, and specialization in $Q^\ell$. Besides roots of unity, special $\mathrm{L}$-values appear in $\lim Q^\ell$ whenever $\bar Y$ admits more than one smoothings. Further techniques are employed and explored beyond known tools in Gromov--Witten theory including (i) the canonical local B model attached to $Y \searrow X$, (ii) existence of semistable reduction of double point type for the smoothing, (iii) the modularity of the extremal function $\mathbb{E} := E^3/\langle E, E, E\rangle^Y$, and (iv) periods integrals of Eisenstein series. Our study provides a geometric framework linking classifications of del Pezzo surfaces, Ramanujan's theta functions, and Zagier's special ODE list via Type II transitions.

math.AG↗
arXiv2025

Remarks on GW/PT under del Pezzo transitions

A projective threefold transition $Y \xrightarrowϕ \bar{Y} \rightsquigarrow X$ is del Pezzo if $ϕ$ contracts a smooth del Pezzo surface to a point. We show that the GW/PT correspondence holds on $Y$ implies that it holds on $X$. In particular, a hypersurface of degree $6$ in $\mathbb{P} (3, 2, 1, 1, 1)$ gives a new example to the correspondence. The main tools are (i) the double point degeneration constructed in arXiv:2508.01374 and (ii) deformations of del Pezzo surfaces into toric surfaces (Proposition 3.12). Applications of the degeneration formulas in GW and PT then reduce the problem to known cases.

math.AG↗
arXiv2025

Gromov--Witten/Pandharipande--Thomas correspondence via conifold transitions

Given a (projective) conifold transition of smooth projective threefolds from $X$ to $Y$, we show that if the Gromov--Witten/Pandharipande--Thomas descendent correspondence holds for the resolution $Y$, then it also holds for the smoothing $X$ with stationary descendent insertions. As applications, we show the correspondence in new cases.

math.AG↗
arXiv2023

The movable cone of Calabi--Yau threefolds in ruled Fano manifolds

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi--Yau threefold in a non-split $(n + 4)$-dimensional $\mathbb{P}^{n}$-ruled Fano manifold of index $n + 1$ and Picard number two. Moreover, all birational minimal models of such Calabi--Yau threefolds are found whose number is finite.

math.AG↗
arXiv2021

The movable cone of certain Calabi-Yau threefolds of Picard number two

We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism.

math.AG↗
arXiv2021

Decorated sheaves and morphisms in tilted hearts

We identify limit stable pairs and stable framed sheaves as epimorphisms and monomorphisms, respectively, in tilts of the standard heart, under suitable conditions. We then identify the moduli spaces with the corresponding Quot spaces, obtaining the projectivity of the Quot spaces in these cases. We also prove a formula in a motivic Hall algebra relating the Quot spaces under a tilt.

math.AG↗
arXiv2020

A note on nodal determinantal hypersurfaces

We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface.

math.AG↗
arXiv2017

On the connectedness of the standard web of Calabi-Yau 3-folds and small transitions

We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call primitive small transitions and study such subclass. More precisely, given a small projective resolution $π: \widehat{X} \rightarrow X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $Def(\widehat{X}) \hookrightarrow Def(X)$ is an isomorphism then $X$ has only ODPs as singularities.

math.AG↗
arXiv2011

Extensions of multiply twisted pluri-canonical forms

Given a projective variety X, a smooth divisor D, and semipositive line bundles (L_1,h_1),,...,(L_m,h_m), we consider the "multiply twisted pluricanonical bundle" F:=m(K_X+D)+L_1+...+L_m on X and F_D:=mK_D+(L_1+...+L_m)|_D. Let I_j be the multiplier ideal sheaves associated to h_j, j=1,...,m. We show that, under a certain conditions on curvature, H^0(D,F_D\otimes I_1I_2...I_m) lies in the image of the restriction map H^0(X,F)->H^0(D,F_D). The format of our result is inspired both by Paun's simplification of Siu's proof of invariance of plurigenera and an earlier similar result due to Demailly. The main ingredient is a modification of Siu-Paun's induction construction and an extension theorem of Ohsawa-Takegoshi type (O-T). We also include a detail proof of O-T. The key feature is that the ideal sheaf we use is the product of the multiplier ideals associated to the singular metrics h_1,...,h_m, which contains the multiplier ideal sheaf of the product of the metrics h_1\otimes...\otimes h_m.

math.AG↗