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We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Theorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P N . Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation.

We study the Seshadri constants on geometrically ruled surfaces. The unstable case is completely solved. Moreover, we give some bounds for the stable case. We apply these results to compute the Seshadri constant of the rational and elliptic ruled surfaces. Both cases are completely determined. The elliptic case provides an interesting picture of how particular is the behavior of the Seshadri constants.MSC (2000): Primary 14C20; secondary, 14J26.

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Ice growth mitigation is a pervasive challenge for multiple industries. In nature, ice-binding proteins (IBPs) demonstrate potent ice growth prevention through ice recrystallization inhibition (IRI). However, IBPs are expensive, difficult to produce in large quantities, and exhibit minimal resilience to nonphysiological environmental stressors, such as pH. For these reasons, researchers have turned to bioinspired polymeric materials that mimic IBP behavior. To date, however, no synthetic polymer has rivaled the ability of native IBPs to display IRI activity at ultralow nanomolar concentrations. In this work, we study the IRI activity of peptides and polypeptides inspired by common ice-binding residues of IBPs to inform the synthesis and characterization of a potent bioinspired polymer that mimics IBP behavior. We show first that the threonine polypeptide (pThr) displays the best IRI activity in phosphate-buffered saline (PBS). Second, we use pThr as a molecular model to synthesize and test a bioinspired polymer, poly­(2-hydroxypropyl methacrylamide) (pHPMA). We show that pHPMA exhibits potent IRI activity in neutral PBS at ultralow concentrations (0.01 mg/mL). pHPMA demonstrates potent IRI activity at low molecular weights (2.3 kDa), with improved activity at higher molecular weights (32.8 kDa). These results substantiate that pHPMA is a robust molecule that mitigates ice crystal growth at concentrations similar to native IBPs.