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We study minimally Terracini finite sets of points in the projective plane and we prove that the sequence of the cardinalities of minimally Terracini sets can have any number of gaps for degree great enough. [ABSTRACT FROM AUTHOR]

A study is made of algebraic curves and surfaces in the flag manifold F = S U (3) / T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of the 4-dimensional sphere S 4 . Deformations of twistor fibers project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1 , 1) . These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1 , 1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F . [ABSTRACT FROM AUTHOR]

We study smooth integral curves of bidegree (1, 1), called smooth conics, in the flag threefold F . The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection F → C P 2 . We give a bound on the maximum number of smooth conics contained in a smooth surface S ⊂ F . Then, we show qualitative properties of algebraic surfaces containing a prescribed number of smooth conics. Last, we study surfaces containing infinitely many twistor fibers. We show that the only smooth cases are surfaces of bidegree (1, 1). Then, for any integer a > 1 , we exhibit a method to construct an integral surface of bidegree (a, a) containing infinitely many twistor fibers. [ABSTRACT FROM AUTHOR]

We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci. [ABSTRACT FROM AUTHOR]

According to Mukai and Iliev, a smooth prime Fano threefold $$X$$ of genus $$9$$ is associated with a surface $$\mathbb{P }(\mathcal{V })$$ , ruled over a smooth plane quartic $$\varGamma $$ , and the derived category of $$\varGamma $$ embeds into that of $$X$$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $$2$$ stable sheaves on $$X$$ with odd determinant. For each $$c_2 \ge 7$$ , we prove that a component of their moduli space $$\mathsf{M}_X(2,1,c_2)$$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $$\varGamma $$ , having enough sections when twisted by $$\mathcal{V }$$ . For $$c_2=7$$ , we prove that $$\mathsf{M}_X(2,1,7)$$ is isomorphic to the blow-up of the Picard variety $$\text{ Pic}^{2}({\varGamma })$$ along the curve parametrizing lines contained in $$X$$ . [ABSTRACT FROM AUTHOR]