Foliations on $$\mathbb {CP}^2$$ CP 2 of degree d with a singular point with Milnor number $$d^2+d+1$$ d 2 + d + 1

Cayley–Bacharach and Singularities of Foliations

“…Each exhibits sufficient conditions on the singular scheme in order that the foliation or distribution becomes uniquely determined by it. The relevance of this type of results has been very clearly explained in Araujo and Correa (2014): in case the general member of a fixed irreducible component of a space of foliations is determined by its singular scheme, the Hilbert scheme becomes a useful tool for describing such component: this is for instance the case of Alcántara (2018), where the study of the Hilbert space of points in CP 2 provides results on the space of foliations of a fixed degree on CP 2 . We don't know of any other results, beside ours, dealing with the problem of existence or not of minimal subschemes of the singular scheme of a foliation.…”

“…from Alcántara (2018, Theorem 3), and let I r be the homogeneous ideal generated by the coefficients of X r . The Milnor number of [X r ] at p = [1, 0, 0] ∈ P 2 is equal to r 2 + r + 1: the singular scheme S r of [X r ] has its support at the single point p and it is a curvilinear scheme (see in Alcántara (2018) the reference on the matter): Indeed, in a local coordinate chart centered at p, the ideal I r p ⊂ O P 2 , p of S r is given by I r p = α r (y, z), z r 2 +r +1 , where…”

New records and noteworthy data of plants, algae and fungi in SE Europe and adjacent regions, 9

“…The anamorphic state was usually observed, while stromata with perithecia were rarely observed. For representative illustrations of X. sicula the reader may consult Moreno et al (2008), Iglesias (2016) andMerino Alcántara (2017). Although characteristic of the remains of O. europaea, the species is also reported to occur on the leaves of Phillyrea latifolia L. (Moreno al.…”