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• 14
  Nov

  CONSTRUCTIONS AND DEFORMATIONS OF P^2XP^2 CALABI--YAU 3-FOLDS IN CODIMENSION 4

  Date: November 14, 2025
  Time: 03:00 PM
  Location: Deans Smart Lab ( SSE Building; Fourth Floor )

  Speaker: Sumayya Mohsin
  Supervisor: Dr Shaheen Nazir
  Student(s): Sumayya Mohsin

  Algebraic geometry deals with the solutions of systems of polynomial equations of varying degrees and in several variables. These solutions (or zero loci) are called algebraic varieties. One of the simplest examples of algebraic varieties is the unit circle, which is the set of zeros of the polynomial equation x^2+y^2 -1=0. We use several techniques, mainly from commutative algebra, to solve geometric problems related to these sets of zeros. In recent years, advances in algebraic geometry have led to its application in studying problems in biology, statistics, computer science, and physics. In statistics and computer science, people are interested in error correction of sample data (e.g., fitting data points to a line). This problem can be expressed as polynomial equations, and finding their solution becomes a problem of algebraic geometry. Similarly, in computational biology, varieties are used to model evolutionary relationships between species. Since algebraic varieties have such widespread applications, it makes sense to classify them. We are particularly interested in a special type of algebraic variety, the Calabi-Yau manifold, due to its significance to theoretical physics, especially string theory. Einstein's theory of general relativity deals with a four-dimensional space-time made up of three spatial dimensions and an extra dimension of time. According to general relativity, gravity is not just an invisible force but the result of massive bodies distorting the very fabric of space-time. This revolutionary idea to unify space, time, matter, and gravity gave rise to an interesting theoretical question: Could there be a vacuum space-time, without matter, which still has some curvature and therefore gravity? The geometric version of this question was asked by the mathematician Eugenio Calabi in the 1950s. He wanted to know whether a compact manifold could have zero Ricci curvature everywhere without being flat (or having zero Riemannian curvature). In terms of physics, this would mean the possibility of a space with no matter but gravity. In 1977, the mathematician Shing-Tung Yau proved the existence of such manifolds in all dimensions, and they have since been known as Calabi-Yau manifolds. We construct and study deformation families of 3-dimensional Calabi-Yau manifolds, with at worst isolated canonical orbifold points, that can be described by a codimension 4 variety in the 7-dimensional weighted projective space P^7(w_i). We obtain such families by using the equations of the Segre embedding of P^2xP^2 in P^8, and we call them Calabi-Yau 3-folds. We first determine a list of 24 Hilbert series that may give rise to a Calabi–Yau threefold in P^2xP^2 format, using a certain bound on the weights of P^7(w_i). Then, for each Hilbert series, we check for the existence of a well-formed and quasismooth Calabi-Yau 3-fold, proving the existence of 23 such families. We compute the Hodge numbers of all P^2xP^2 format varieties and of their crepant resolutions. One family displays Hodge numbers that do not seem to appear in the famously known existing databases of Hodge pairs of Calabi-Yau 3-folds, thereby expanding the current Calabi-Yau landscape. We then study their Hilbert schemes in two ways: (i) we perform Tom and Jerry degenerations (two Pfaffian deformation formats), and (ii) we compare their Hilbert series with those of known low-codimension Calabi-Yau threefolds. Alongside other interesting phenomena, we obtain a family with five distinct Tom or Jerry deformation families, a level of richness not observed in the Q-Fano 3-fold case.

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