Nodal surfaces with obstructed deformations

In this text we show that the deformation space of a nodal surface $X$ of degree $d$ is smooth and of the expected dimension if $d\leq 7$ or $d\geq 8$ and $X$ has at most $4d-5$ nodes. (The case $d\leq 7$ was previously covered by Alexandru Dimca by using different techniques.) For $d\geq 8$ we give explicit examples of nodal surfaces with $4d-4$ nodes, for which the tangent space to the deformation space has larger dimension than expected. We give a short discussion on the shape of the deformation space of surfaces of the form $f_1f_2+f_3^2f_4$, where $f_1$ is a linear form.Comment: v2: Added a reference to a similar result by Alexandru Dimca and a discussion on the difference between Dimca's result and ours v3: Expanded several argument

Maximal families of nodal varieties with defect

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large.Comment: v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different pape

Extremal elliptic surfaces & Infinitesimal Torelli

We describe in terms of the j-invariant all elliptic surfaces pi: X -> C with a section, such that h^{1,1}(X)=rank NS(X) and the Mordell-Weil group of pi is finite. We use this to give a complete solution to infinitesimal Torelli for elliptic surfaces with a section over P^1.Comment: 16 pages; 3rd version; small changes to the third and fourth sectio

Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Let $X_\lambda$ and $X_\lambda'$ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249]