[Paper Review] Q-Gorenstein deformations of non-normal surfaces

Nikolaos Tziolas|ArXiv.org|Apr 11, 2005

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper characterizes the sheaf $ T^{1}_{qG}(H) $ of $ \mathbb{Q} $-Gorenstein deformations for non-normal surfaces $ H $ with semi-log-canonical (slc) singularities along a proper curve $ \Delta $, showing its divisorial support is exactly $ \Delta $. It proves that for each irreducible component $ C \subset \Delta $, the locally free part of $ T^{1}_{qG}(H) \otimes \mathcal{O}_C $ is a line bundle and derives a degree formula involving $ \tilde{C}^2 $ and invariants of singularities. The work provides criteria for $ \Delta \subset H $ to admit $ \mathbb{Q} $-Gorenstein terminal smoothings.

ABSTRACT

Revised Version. An example of a locally smoothable stable surface that does not have a global smoothing has been added.

Motivation & Objective

• To understand the global $ \mathbb{Q} $-Gorenstein deformation theory of non-normal surface germs $ \Delta \subset H $ with semi-log-canonical (slc) singularities.
• To determine when local $ \mathbb{Q} $-Gorenstein smoothings of such surface germs extend to global smoothings.
• To describe the sheaf $ T^{1}_{qG}(H) $ of $ \mathbb{Q} $-Gorenstein deformations and identify its support and structure on components of $ \Delta $.
• To derive a degree formula for the line bundle component of $ T^{1}_{qG}(H) \otimes \mathcal{O}_C $ on each irreducible component $ C \subset \Delta $.
• To establish criteria for the existence of $ \mathbb{Q} $-Gorenstein terminal smoothings of $ \Delta \subset H $.

Proposed method

• Define the global and local $ \mathbb{Q} $-Gorenstein deformation functors $ \mathrm{Def}^{qG}(H) $ and $ \mathrm{Def}^{qG}_{\text{loc}}(H) $, and study the natural transformation between them.
• Characterize the tangent space of $ \mathrm{Def}^{qG}_{\text{loc}}(H) $ as $ H^0(T^{1}_{qG}(H)) $, where $ T^{1}_{qG}(H) \subset T^1(H) $ is the sheaf of local $ \mathbb{Q} $-Gorenstein deformations.
• Show that the divisorial part of the support of $ T^{1}_{qG}(H) $ is exactly $ \Delta $, the curve along which $ H $ is non-normal.
• For each irreducible component $ C \subset \Delta $, prove that $ T^{1}_{qG}(H) \otimes \mathcal{O}_C $ has a locally free part that is a line bundle.
• Derive a degree formula for this line bundle in terms of $ \tilde{C}^2 $, the self-intersection of the strict transform under normalization $ \pi: \tilde{H} \to H $, and analytic invariants of singularities.
• Use minimal resolutions of $ \tilde{H} $ to compute invariants such as $ \hat{F}_P^2 $, $ \hat{F}_{1,Q}^2 $, $ \hat{F}_{2,Q}^2 $, and intersection numbers to verify the degree formula.

Experimental results

Research questions

• RQ1 What is the structure of the sheaf $ T^{1}_{qG}(H) $ of $ \mathbb{Q} $-Gorenstein deformations for a non-normal surface germ $ \Delta \subset H $ with slc singularities?
• RQ2 When do local $ \mathbb{Q} $-Gorenstein smoothings of such a surface germ extend to a global smoothing?
• RQ3 What is the degree of the line bundle that arises as the locally free part of $ T^{1}_{qG}(H) \otimes \mathcal{O}_C $ for each irreducible component $ C \subset \Delta $?
• RQ4 Which singularities of class qG (e.g., degenerate cusps, $ \mathbb{Z}_n $-quotients) allow for $ \mathbb{Q} $-Gorenstein terminal smoothings?
• RQ5 How do the invariants $ \tilde{C}^2 $, $ \beta_3(P) $, $ \beta_4(Q) $, and $ \delta_3(P) $, $ \delta_4(Q) $ relate in the degree formula for the line bundle on $ C $?

Key findings

• The divisorial part of the support of $ T^{1}_{qG}(H) $ is exactly $ \Delta $, the curve along which $ H $ is non-normal.
• For each irreducible component $ C \subset \Delta $, the sheaf $ T^{1}_{qG}(H) \otimes \mathcal{O}_C $ has a locally free part that is a line bundle.
• The degree of this line bundle is given by a formula involving $ \tilde{C}^2 $ and invariants of the singularities, specifically $ \sum_P \beta_3(P) + \sum_Q \beta_4(Q) + \sum_R \beta_4'(R) $, where the sums are over degenerate cusps of types $ T^3_{p,q} $, $ T^4_{p,q,r} $, and $ T^4_{2,q,r} $.
• For a degenerate cusp $ P \in U_3 $ of type $ T^3_{p,q} $, the contribution to the degree is $ \beta_3(P) = -1 + \frac{1}{p} + \frac{1}{q} $, and the correction term $ \delta_3(P) = \frac{(p+q)^2}{pq(p+q+pq)} $.
• For a degenerate cusp $ Q \in W_4 $ of type $ T^4_{p,q,r} $, the contribution is $ \beta_4(Q) = -1 + \frac{1}{p-2} + \frac{1}{q-2} $, and $ \delta_4(Q) = \frac{1}{p-2} + \frac{1}{q-2} - \frac{r(p+q)-4}{rpq - p - q} $.
• The degree formula leads to $ d = \tilde{C}^2 + p + 2c_1 + 2c_2 + \sum_{P \in U_3} \alpha_3(P) + \sum_{Q \in U_4} \alpha_4(Q) $, where $ \alpha_3(P) = 1 + \frac{p+q}{pq + p + q} $ and $ \alpha_4(Q) = \frac{r(p+q)-4}{rpq - p - q} $, with limits taken at infinity for infinite parameters.