arXiv subjects

Julie Rana

Publications and source records attributed to Julie Rana.

8 recordsLinked to original sources

arXiv2024

Optimal bounds for many T-singularities in stable surfaces

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the vast increase in combinatorial complexity as the number of singularities grows. We find that certain combinatorial configurations lead to relatively high bounds. We classify all such configurations, and show that their non-existence gives a strong and optimal bound. As an application, we work out in detail the case of two singularities.

math.AG↗

arXiv2022

Standard stable Horikawa surfaces

We consider the stable compactification $\bar {\mathfrak H}$ of the moduli space of Horikawa surfaces with $K_X^2 = 2p_g(X) -4$. When $K_X^2 =8\ell$ we show that the closures of the two components $\mathfrak H^{\mathrm I}$ and $\mathfrak H^{\mathrm {II}}$ of the Gieseker moduli space intersect, for $\ell>2$ in a divisor parametrising explicitly described semi-smooth surfaces. With growing $K_X^2$ we find an increasing number of generically non-reduced irreducible components in the same connected component of the moduli space of stable surfaces.

math.AG↗

arXiv2021

On T-divisors and intersections in $\overline{M}_{1,3}$

The moduli space of stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$ has at least two irreducible components that contain surfaces with T-singularities. We show that the two known components intersect transversally in a divisor. Moreover, we exhibit other new boundary divisors and study how they intersect one another.

math.AG↗

arXiv2020

I-surfaces with one T-singularity

We classify normal stable surfaces with $K_X^2 = 1$, $p_g = 2$ and $q=0$ with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli space of stable surfaces $\overline\gothM_{1,3}$.

math.AG↗

arXiv2020

Optimal bounds for T-singularities in stable surfaces

We explicitly bound T-singularities on normal projective surfaces $W$ with one singularity, and $K_W$ ample. This bound depends only on $K_W^2$, and it is optimal when $W$ is not rational. We classify and realize surfaces attaining the bound for each Kodaira dimension of the minimal resolution of $W$. This answers effectiveness of bounds (see [Alexeev94], [Alexeev-Mori04], [Lee99]) for those surfaces.

math.AG↗

arXiv2017

Equations of $\,\overline{M}_{0,n}$

Following work of Keel and Tevelev, we give explicit polynomials in the Cox ring of $\mathbb{P}^1\times\cdots\times\mathbb{P}^{n-3}$ that, conjecturally, determine $\overline{M}_{0,n}$ as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for $n=5, 6, 7, 8$. For $n \leq 6$ we give a cohomological proof that these polynomials realize $\overline{M}_{0,n}$ as a projective variety, embedded in $\mathbb{P}^{(n-2)!-1}$ by the complete log canonical linear system.

math.AG↗

arXiv2016

The Craighero-Gattazzo surface is simply-connected

We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod p technique and deformation theory.

math.AG↗

arXiv2014

A boundary divisor in the moduli space of stable quintic surfaces

We give a bound on which singularities may appear on Kollár--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with $K^2=χ=5$) whose unique non Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space $\overline{\mathcal{M}}_{5,5}$ corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

math.AG↗