Introduction to Toric Varieties. (AM-131)

Fulton, William

doi:10.1515/9781400882526

Cited by 2,118 publications

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“…Indeed for a projective toric compactification X of (C * ) n such that the closureZ * of Z * in X is smooth (see [13,32] etc. ), the varietyZ * is smooth projective and hence there exists a perfect pairing…”

Section: B C (X)mentioning

confidence: 99%

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Esterov

Takeuchi

2011

International Mathematics Research Notices

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Section: B C (X)mentioning

confidence: 99%

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Esterov

Takeuchi

2011

International Mathematics Research Notices

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“…The quotient is a toric surface with a rational singularity. See [Ful93], Section 2.6, for a complete discription. Let R be the local ring at the singular point, and set X = Spec(R).…”

Section: Ementioning

confidence: 99%

Jumping numbers on algebraic surfaces with rational singularities

Tucker

2009

Trans. Amer. Math. Soc.

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Abstract. In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

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“…The underlying real torus T m action on the symplectic manifold (M P , ω P ) is Hamiltonian with moment map µ P : M P → P fibering M P as a singular torus bundle over the polytope P (see, e.g., [Fu,§4.2]). The volume form on M P can be written in terms of the moment map: 1 m!…”

Section: ])mentioning

confidence: 99%

Random polynomials with prescribed Newton polytope

Shiffman

Zelditch

2003

J. Amer. Math. Soc.

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The Newton polytope P f P_f of a polynomial f f is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in ( C ∗ ) m (\mathbb {C}^*)^m of a system of m m polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in ( C ∗ ) m (\mathbb {C}^*)^m for these distributions. Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree ≤ p \leq p in m m complex variables with its usual SU ( m + 1 ) (m+1) -invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope P P . We then determine the asymptotics of the conditional expectation E | N P ( Z f 1 , … , f k ) \mathbf {E}_{|N P}(Z_{f_1, \dots , f_k}) of simultaneous zeros of k k polynomials with Newton polytope N P NP as N → ∞ N \to \infty . When P = Σ P = \Sigma , the unit simplex, it is clear that the expected zero distributions E | N Σ ( Z f 1 , … , f k ) \mathbf {E}_{|N\Sigma }(Z_{f_1, \dots , f_k}) are uniform relative to the Fubini-Study form. For a convex polytope P ⊂ p Σ P\subset p\Sigma , we show that there is an allowed region on which N − k E | N P ( Z f 1 , … , f k ) N^{-k}\mathbf {E}_{|N P}(Z_{f_1, \dots , f_k}) is asymptotically uniform as the scaling factor N → ∞ N\to \infty . However, the zeros have an exotic distribution in the complementary forbidden region and when k = m k = m (the case of the Bernstein-Kouchnirenko Theorem), the expected percentage of simultaneous zeros in the forbidden region approaches 0 as N → ∞ N\to \infty .

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Introduction to Toric Varieties. (AM-131)

Cited by 2,118 publications

References 0 publications

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

Jumping numbers on algebraic surfaces with rational singularities

Random polynomials with prescribed Newton polytope

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