Alessandro Oneto

On the real rank of monomials

In this paper we study the real rank of monomials and we give an upper bound for it. We show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one.

MONOMIALS AS SUMS OF k th -POWERS OF FORMS

Motivated by recent results on the Waring problem for polynomial rings [4] and representation of monomial as sum of powers of linear forms [3], we consider the problem of presenting monomials of degree kd as sums of kth-powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k = 3 case for monomials in two and three variables.

On a class of power ideals

In this paper we study the class of power ideals generated by the k(n) forms (x(0) + xi(g1) x(1) + ... + xi(gn) x(n))((k-1)d) where xi is a fixed primitive kth-root of unity and 0 2. Via Macaulay ...

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f. In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with 13

Mirror symmetry and the classification of orbifold del Pezzo surfaces

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On the quantum periods of del Pezzo surfaces with

(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.