I am mainly interested in complex algebraic geometry mostly Hodge theory and toric varieties. I study how to generalize some classical results about the Noether-Lefschetz loci to projective simplicial toric varieties and how to go deeper in the classical case. The Noether-Lefschetz loci is the locus parameterizing the hypersurfaces for which the Hodge conjecture is non-trivial. In addition, I am studying the connection between the Noether-Lefschetz theory in toric varieties and Mori dream spaces. I am also working in non-standard Artinian Gorenstein algebras motivated by the primitive cohomology of a general hypersurface in a projective toric orbifold. Recently, I have been exploring an extended logarithmic tangent sheaf in toric geometry for hyperplane arrangements and its correspondence with foliations.
Works in Progress
Laface, A., Massarenti, A., and Montoya, W. D. (202X). Mori dream K3 surfaces with Picard rank 2 in weighted projective 3-folds.
Bruzzo, U., Grassi, A., and Montoya, W. D. (202X). Explicit Noether-Lefschetz theorem in toric 3-folds.
Montoya, W. D. (202X). 𝜙-Lexsegment ideals and Macaulay’s theorem.
Publications and Preprints
(reverse chronological order)
Bruzzo, U., Gondim, R., Holanda, R. & Montoya, W. D. (2024). Cox Gorenstein Algebras. arXiv math. AC. 2407.17811. https://doi.org/10.48550/arXiv.2407.17811.
Bruzzo, U. & Montoya, W. D. (2024). An asymptotic description of the Noether Lefschetz components in toric varieties. arXiv:1905.01570 [math.AG]. To appear in Revista Matemática Iberoamericana. https://doi.org/10.48550/arXiv.1905.01570.
Faenzi, D., Jardim, M. & Montoya, W. D. (2024). Logarithmic vector fields and foliations on toric varieties. hal-04669484, version 1. https://hal.science/hal-04669484.
Montoya, W. D. (2024). Some open problems in Noether Lefschetz theory in toric varieties. Matemática Contemporânea, 60, 163-184. http://doi.org/10.21711/231766362024/rmc608.
Bruzzo, U. & Montoya, W. D. (2023). Deformation of pairs and Noether-Lefschetz loci in toric varieties. European Journal of Mathematics, 9(4), 108. https://doi.org/10.1007/s40879-023-00702-4.
Montoya, W. D. (2023). A weak (k,k)-Lefschetz theorem for projective toric orbifolds. arXiv:2302.03803 [math.AG]. https://doi.org/10.48550/arXiv.2302.03803.
Bruzzo, U. & Montoya, W. D. (2021). Codimension bounds of the Noether Lefschetz components for toric varieties. European Journal of Mathematics, 1-9. https://doi.org/10.1007/s40879-021-00461-0.
Bruzzo, U. & Montoya, W. D. (2021). On the Hodge conjecture for quasi-smooth intersections in toric varieties. São Paulo Journal of Mathematical Sciences, 15(2), 682-694. https://doi.org/10.1007/s40863-021-00247-y.