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Deniz Genlik

Publications and source records attributed to Deniz Genlik.

5 recordsLinked to original sources
arXiv2026

Polytopes of Effective Boundary Expressions of Divisors on $\overline{M}_{0,n}$

For a divisor on $\overline{M}_{0,n}$, we introduce the polytope of its effective boundary expressions. We establish structural properties of these polytopes under the forgetful maps of $\overline{M}_{0,n}$ forgetting marked points, and give equivalent graph-theoretic descriptions. We compute these polytopes for several families of divisors. For psi-classes and their pullbacks by forgetful maps, we show that the polytopes are unimodular simplices. For the log-canonical class and its modifications by psi-classes, we prove that the nonnegative parts of the corresponding polytopes recover spanning forest polytopes and the subtour elimination (Held--Karp relaxation) polytope of the symmetric traveling salesman problem. As an application, we obtain a Minkowski-like decomposition of the subtour elimination polytope into simplices. Finally, for symmetric level-one $\mathfrak{sl}_p$ conformal block divisors, we show that the defining inequalities are local Tur\'an bounds and the $0/1$-points are balanced Tur\'an graphs. Moreover, for $p=2$ and $p=n/2$, these polytopes recover the perfect matching and fractional perfect matching polytopes.

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arXiv2024

Higher Genus Gromov-Witten Theory of C^n/Z_n I: Holomorphic Anomaly Equations

We study the structure of higher genus Gromov-Witten theory of the quotient stack $[\mathbb{C}^n/\mathbb{Z}_n]$. We prove holomorphic anomaly equations for $[\mathbb{C}^n/\mathbb{Z}_n]$, generalizing previous results of Lho-Pandharipande arXiv:1804.03168 for the case of $[\mathbb{C}^3/\mathbb{Z}_3]$ and ours arXiv:2211.15878 for the case $[\mathbb{C}^5/\mathbb{Z}_5]$ to arbitrary $n\geq{3}$.

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arXiv2023

Higher Genus Gromov-Witten Theory of C^n/Z_n II: Crepant Resolution Correspondence

We study the structure of the higher genus Gromov-Witten theory of the total space $K\mathbb{P}^{n-1}$ of the canonical bundle of the projective space $\mathbb{P}^{n-1}$. We prove the finite generation property for the Gromov-Witten potential of $K\mathbb{P}^{n-1}$ by working out the details of its cohomological field theory (CohFT). More precisely, we prove that the Gromov-Witten potential of $K\mathbb{P}^{n-1}$ lies in an explicit polynomial ring using the Givental-Teleman classification of the semisimple CohFTs. In arXiv:2301.08389, we carried out a parallel study for $[\mathbb{C}^n/\mathbb{Z}_n]$ and proved that the Gromov-Witten potential of $[\mathbb{C}^n/\mathbb{Z}_n]$ lies in a similar polynomial ring. The main result of this paper is a crepant resolution correspondence for higher genus Gromov-Witten theories of $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$, which is proved by establishing an isomorphism between the polynomial rings associated to $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$. This paper generalizes the works of Lho-Pandharipande arXiv:1804.03168 for the case of $[\mathbb{C}^3/\mathbb{Z}_3]$ and Lho arXiv:2211.15878 for the case $[\mathbb{C}^5/\mathbb{Z}_5]$ to arbitrary $n\geq 3$.

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