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Joe Harris

Publications and source records attributed to Joe Harris.

13 recordsLinked to original sources
arXiv2020

Biominerals with Texture Gradients are Functionally Graded Bioceramics Toughened by Stress Delocalization

Biomineralizing organisms are widely noted and extensively studied due to their ability to generate structures exhibiting exceptional crystallographic control. Primarily, it is the organisms, such as sea-urchins or bivalves, that generate nearly single-crystalline biocrystals that have attracted attention. In contrast, biomineralizing organisms with seemingly disordered polycrystalline bio-armor have been left relatively unstudied. However, the crystalline ordering in the black-lipped pearl oyster, Pinctada margaritifera, reveals that biominerals with varying crystal textures are an unrecognized class of functionally graded materials. Changing crystal textures inevitably cause a variation in Young modulus due to the orientation-dependent mechanical properties of crystals. The case of Pinctada margaritifera demonstrates that bioceramics with such crystallographical gradients are toughened by stress delocalization and reduced stress intensity factors, outperforming non-graded counterparts. These findings suggest that a multitude of biominerals, which are perceived as poorly ordered because of their polycrystallinity and changing crystal textures, may be considered as graded materials with hitherto unidentified emergent mechanical properties. The underlying design principle is remarkably simple and applicable to a wide range of crystalline material classes and may thus serve as a blueprint for future bioinspired functional materials.

cond-mat.mtrl-sci↗
arXiv2008

Powers of Ideals and Fibers of Morphisms

Let X\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of ϕ, considered as subschemes of \PP^n. From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+εfor some non-negative integers d, ε, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of ε: it is one less than the maximum regularity of a fiber of the morphism associated to I. These formulas have strong consequences for ideals generated by generic forms.

math.AG↗
arXiv2002

Rational curves on hypersurfaces of low degree, II

This is a continuation of "Rational curves on hypersurfaces of low degree", math.AG/0203088. We prove that if d^2+d+1 < n and d > 2, then for a general hypersurface X_d in P^n of degree d, for each degree e the space of rational curves of degree e on X is itself a rationally connected variety.

math.AG↗
arXiv2002

A note on Hurwitz schemes of covers of a positive genus curve

We prove the irreducibility of the space parametrizing branched covers of a fixed Riemann surface $B$ of degree $d$, with at least 2d branch points, and with monodromy group equal to $S_d$. The result is classical for $g(B)=0$. The result is well-known for $g(B) > 0$, but we could find no reference.

math.AG↗
arXiv2002

Rational curves on hypersurfaces of low degree

Let n > 2 and let d < (n+1)/2. We prove that for a general hypersurface X of degree d in P^n, all the genus 0 Kontsevich moduli spaces M_{0,n}(X,e) are irreducible, reduced, local complete intersection stacks of the expected dimension.

math.AG↗
arXiv2002

Abel-Jacobi maps associated to smooth cubic threefolds

We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by analyzing vector bundles on cubic threefolds. Our proofs use residuation rather than vector bundles.

math.AG↗
arXiv2002

Curves of small degree on cubic threefolds

We describe the Hilbert schemes parametrizing curves on a cubic threefold of degree at most 5. In a forthcoming paper, we use this description to give a new proof and extension of a theorem of Iliev, Markushevich and Tikhimirov.

math.AG↗
arXiv1998

Rational Points on Quartics

Let $S \subset ¶^n$ be a smooth quartic hypersurface defined over a number field $K$. If $n \ge 4$, then for some finite extension $K'$ of $K$ the set $S(K')$ of $K'$-rational points of $S$ is Zariski dense.

math.AG↗
arXiv1996

Counting plane curves of any genus

We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.

alg-geom↗